ROMANSY 23 - Robot Design, Dynamics and Control: Proceedings of the 23rd CISM IFToMM Symposium [1st ed.] 9783030583798, 9783030583804

Citation preview

CISM International Centre for Mechanical Sciences 601 Courses and Lectures

Gentiane Venture Jorge Solis Yukio Takeda Atsushi Konno   Editors

ROMANSY 23 Robot Design, Dynamics and Control Proceedings of the 23rd CISM IFToMM Symposium International Centre for Mechanical Sciences

CISM International Centre for Mechanical Sciences Courses and Lectures Volume 601

Managing Editor Paolo Serafini, CISM—International Centre for Mechanical Sciences, Udine, Italy Series Editors Elisabeth Guazzelli, IUSTI UMR 7343, Aix-Marseille Université, Marseille, France Franz G. Rammerstorfer, Institut für Leichtbau und Struktur-Biomechanik, TU Wien, Vienna, Wien, Austria Wolfgang A. Wall, Institute for Computational Mechanics, Technical University Munich, Munich, Bayern, Germany Bernhard Schrefler, CISM—International Centre for Mechanical Sciences, Udine, Italy

For more than 40 years the book series edited by CISM, “International Centre for Mechanical Sciences: Courses and Lectures”, has presented groundbreaking developments in mechanics and computational engineering methods. It covers such fields as solid and fluid mechanics, mechanics of materials, micro- and nanomechanics, biomechanics, and mechatronics. The papers are written by international authorities in the field. The books are at graduate level but may include some introductory material.

More information about this series at http://www.springer.com/series/76

Gentiane Venture Jorge Solis Yukio Takeda Atsushi Konno •

•

•

Editors

ROMANSY 23 - Robot Design, Dynamics and Control Proceedings of the 23rd CISM IFToMM Symposium

123

Editors Gentiane Venture Tokyo University of Agriculture and Technology Tokyo, Japan Yukio Takeda Tokyo Institute of Technology Tokyo, Japan

Jorge Solis Karlstad University Karlstad, Sweden Atsushi Konno Hokkaido University Sapporo, Japan

ISSN 0254-1971 ISSN 2309-3706 (electronic) CISM International Centre for Mechanical Sciences ISBN 978-3-030-58379-8 ISBN 978-3-030-58380-4 (eBook) https://doi.org/10.1007/978-3-030-58380-4 © CISM International Centre for Mechanical Sciences 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

It is a great pleasure to present to you the 23rd CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators. This volume highlights the latest advances, innovations, and applications in robotics, as presented by leading international researchers and engineers at the conference held online during September 20–24, 2020. ROMANSY symposium is the first established conference that emphasizes the theory and research of robotics, rather than the industrial aspect. Thanks to the active participation of researchers from many countries, the symposium is held bi-annually and is playing a vital role in the development of the theory and practice of robotics, as well as of mechanical sciences. ROMANSY 2020 is the 23rd event in a series that started in 1973. It was also the first topic conference of IFToMM (International Federation for the Promotion of Mechanism and Machine Science), and it was directed not only to the IFToMM community. Even though the conference couldn’t be held in person, ROMANSY 2020 is faithful to this tradition. We would like to express our deepest gratitude for the enthusiasm and dedication shown by the ROMANSY committee that supported the preparation of the proceedings and all the authors of the presented papers. Gentiane Venture Jorge Solis Program Chairs Yukio Takeda Atsushi Konno General Chairs

v

Contents

Active Visualization of Non-destructive Inspection for Metal Using Terahertz Camera and Light Source . . . . . . . . . . . . . . . . . . . . . . . . . . . Shunsuke Yamada, Teppei Tsujita, Masahiro Kurosaki, Tetsuo Tomizawa, Yutaka Sakuma, and Ryosuke Eto Model-Based Dynamic Human Tracking and Reconstruction During Dynamic SLAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huayan Zhang, Tianwei Zhang, and Lei Zhang Development of Robust Ridge Detection Method and Control System for Autonomous Navigation of Mobile Robot in Agricultural Farm . . . . Shunsuke Fujita, Takanori Emaru, Ankit A. Ravankar, and Yukinori Kobayashi Continuous Jumping Control Based on Virtual Model Control for a One-Leg Robot Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Libo Meng, Marco Ceccarelli, Zhangguo Yu, Xuechao Chen, Gao Huang, and Qiang Huang

1

8

16

24

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System for Biped Humanoid Robot Focusing on Human Running Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hideki Mizukami, Takuya Otani, Juri Shimizu, Kenji Hashimoto, Masanori Sakaguchi, Yasuo Kawakami, Hun-ok Lim, and Atsuo Takanishi

34

Maximal Output Admissible Set of Foot Position Control in Humanoid Walking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ko Yamamoto, Ryo Yanase, and Yoshihiko Nakamura

43

Control System Design for Human Assisting Robot . . . . . . . . . . . . . . . . Teresa Zielinska and Michele Tartari

52

vii

viii

Contents

Development of a Switchable Wearable Robot for Rehabilitation After Surgery of Knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koji Makino, Teppei Ogura, Masahiro Nakamura, and Hidetsugu Terada Design and Testing of BIT Flying Robot . . . . . . . . . . . . . . . . . . . . . . . . Yunqi Liu, Long Li, Marco Ceccarelli, Hui Li, Qiang Huang, and Xiang Wang Surgical Skill Analysis Based on the Way of Grasping Organs with Forceps in Dissection Procedure of Laparoscopic Surgery . . . . . . . Koki Ebina, Takashige Abe, Shunsuke Komizunai, Teppei Tsujita, Kazuya Sase, Xiaoshuai Chen, Madoka Higuchi, Jun Furumido, Naoya Iwahara, Yo Kurashima, Nobuo Shinohara, and Atsushi Konno Mechanism and Control of Powered Prosthesis with Bi-articular Muscle-Type Hydraulic Bilateral Servo Actuator . . . . . . . . . . . . . . . . . . Takanori Higashihara, Toru Oshima, Takumi Tamamoto, Kengo Ohnishi, Ken’ichi Koyanagi, and Yukio Saito Development of a Remote-Controlled Drone System by Using Only Eye Movements for Bedridden Patients . . . . . . . . . . . . . . . . . . . . . . . . . Atsunori Kogawa, Moeko Onda, Yoshihiro Kai, Tetsuya Tanioka, Yuko Yasuhara, and Hirokazu Ito

60

68

76

84

92

Development of a Climbing-Robot for Spruce Pruning: Preliminary Design and First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Giovanni Carabin, Davide Emanuelli, Raimondo Gallo, Fabrizio Mazzetto, and Renato Vidoni A Suspended Cable-Driven Parallel Robot for Human-Cooperative Object Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Yusuke Sugahara, Guangcan Chen, Nanato Atsumi, Daisuke Matsuura, Yukio Takeda, Ryo Mizutani, and Ryuta Katamura Automatic Planning of Psychologically Less-Stressful Trajectories in Collaborative Workstations: An Integrated Toolbox for Unskilled Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Rafael A. Rojas, Manuel A. Ruiz Garcia, Luca Gualtieri, Erich Wehrle, Erwin Rauch, and Renato Vidoni A New Method of Climbing Downstairs by Changing Layers of Gears of Planetary Wheels for Wheelchair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Tian-ci Jiang and Eiichiro Tanaka Design and Analysis of Cable-Driven Parallel Robot CaRISA: A Cable Robot for Inspecting and Scanning Artwork . . . . . . . . . . . . . . 136 Philipp Tempel, Matthias Alfeld, and Volkert van der Wijk

Contents

ix

A Low Cost Introductory Platform for Advanced Robotic Control . . . . 145 Bin Wei Experimental Study Regarding Needle Deflection in Robotic Assisted Brachytherapy of Hepatocellular Carcinoma . . . . . . . . . . . . . . . . . . . . . 154 Paul Tucan, Nicolae Plitea, Bogdan Gherman, Nadim al Hajjar, Corina Radu, Calin Vaida, and Doina Pisla Experimental Study of Force Transmission in 4-DOF Parallel Manipulator and Its Educational Applications . . . . . . . . . . . . . . . . . . . . 162 Pavel Laryushkin, Elizaveta Pukhova, and Ksenia Erastova Dynamic Model of Servo Mechanical Press . . . . . . . . . . . . . . . . . . . . . . 170 Assylbek Jomartov, Amandyk Tuleshov, Nutpulla Jamalov, Askar Seydakhmet, Sayat Ibrayev, Moldyr Kuatova, Ablay Kaimov, Yerbol Temirbekov, and Bayandy Bostanov A Decentralized Structure for the Digital Shadows of Internet of Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Amir Shahidi, Mathias Hüsing, and Burkhard Corves Motion Trajectory Optimization of an Assistive Device During Stairs Ascending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Bo-Rong Yang, Shuai-Hong Yu, Kai Pang, Hee-Hyol Lee, and Eiichiro Tanaka Autonomous Flight of a Quad Tilt-Rotor UAV at Constant Altitude . . . 195 Satoko Abiko and Tomohiro Harada Control-Based Design of a DELTA Robot . . . . . . . . . . . . . . . . . . . . . . . 204 Minglei Zhu, Abdelhamid Chriette, and Sébastien Briot Minimizing the Energy Consumption of a Delta Robot by Exploiting the Natural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Rafael Balderas Hill, Sébastien Briot, Abdelhamid Chriette, and Philippe Martinet Preliminary Design and Modeling of a Robot for Pipe Navigation with a Novel Wheel-Leg Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Carl A. Nelson Sensitivity Analysis of Cable Actuations for Moving a Tensegrity Mechanism Along a Specified Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Pramod Kumar Malik, Keshab Patra, and Anirban Guha An Approach to Motion Task-Oriented, Computer-Aided Design of Origami-Inspired Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Judith U. Merz, Felix J. Reimer, Mathias Hüsing, and Burkhard Corves

x

Contents

Asymmetric Spatial Beams with Symmetric Kinetostatic Behaviour . . . 247 Ali Amoozandeh Nobaveh, Giuseppe Radaelli, and Just L. Herder A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Naoto Kimura and Nobuyuki Iwatsuki Walking Robot Leg Design Based on Translatory Straight-Line Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Sayat Ibrayev, Nutpulla Jamalov, Amandyk Tuleshov, Assylbek Jomartov, Aidos Ibrayev, Aziz Kamal, Arman Ibrayeva, and Kuatbay Bissembayev Developing a Flexible Segment Unit for Redundant-DOF Manipulator Using Bending Type Pneumatic Artificial Muscle . . . . . . . . . . . . . . . . . . 272 Hiroki Tomori, Tomohiro Koyama, Hiromitsu Nishikata, Akinori Hayasaka, and Ikumi Suzuki Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Vu Linh Nguyen, Chin-Hsing Kuo, and Chyi-Yeu Lin Design of the Mobile Robot Agri.q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Paride Cavallone, Carmen Visconte, Luca Carbonari, Andrea Botta, and Giuseppe Quaglia Kinematic Design of an Adjustable Foot Motion Generator for Gait Rehabilitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Chanatip Thongsookmark, Agnes Beckermann, Mathias Hüsing, and Yukio Takeda Design and Construction of the DragonBall . . . . . . . . . . . . . . . . . . . . . . 305 Bir Bikram Dey and Michael Jenkin Wire-Pulling Mechanism with Embedded Soft Tubes for Robot Tongue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Nobutsuna Endo Kineto-static Analysis of a Compact Wrist Rehabilitation Robot Including the Effect of Human Soft Tissue to Compensate for Joint Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Ying-Chi Liu and Yukio Takeda A Mobile Robot Which Locomotes on Walls to Interact with Rodents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Soichi Yamada, Keitaro Ishibashi, Hiroya Yokoyama, Jiei Yanagi, Atsuo Takanishi, and Hiroyuki Ishii

Contents

xi

Development of Small Robot with Inline Archimedean Screw Mechanism that Can Move Through Wetlands . . . . . . . . . . . . . . . . . . . 338 Ko Matsuhiro, Katsuaki Tanaka, Shou Inoue, Tingting Zhong, Kazuki Kida, Yusuke Sugahara, Atsuo Takanishi, and Hiroyuki Ishii Static Force Analysis of an Omnidirectional Mobile Robot with Wheels Connected by Passive Sliding Joints . . . . . . . . . . . . . . . . . . 347 Tatsuro Terakawa and Masaharu Komori Development of a Trapezoidal Leaf Spring for a Small and Light Variable Joint Stiffness Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Hiroki Mineshita, Takuya Otani, Kenji Hashimoto, Masanori Sakaguchi, Yasuo Kawakami, Hun-ok Lim, and Atsuo Takanishi Connecting MATLAB/Octave to Perceptual, Cognitive and Control Components for the Development of Intelligent Robotic Systems . . . . . . 364 Enrique Coronado, Liz Rincon, and Gentiane Venture Balancing of Planar 5R Symmetrical Parallel Manipulators Taking into Account the Varying Payload . . . . . . . . . . . . . . . . . . . . . . . 372 Jing Geng and Vigen Arakelian A Wheeled Vehicle Driven by a Savonius–Magnus Wind Turbine . . . . . 380 Marat Dosaev, Margarita Ishkhanyan, Liubov Klimina, Anna Masterova, and Yury Selyutskiy Development of an Off-board Vision-Based Control for a Micro Aerial Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Jorge Solis, Christoffer Karlsson, and Kristoffer Richardsson Stiffness Optimization of Delta Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Christian Mirz, Olaf Uzsynski, Jorge Angeles, Yukio Takeda, and Burkhard Corves Singularity Free Mode Changes of a Redundantly Driven Two Limbs Six-Dof Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Takashi Harada and Yuta Kunishige Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Arati Ajay Bhattu and Salil Kulkarni Proposition of On-Line Velocity Scaling Algorithm for Task Space Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Marek Wojtyra and Łukasz Woliński Stability Analysis and Reconfiguration Strategy for Multi-agent D-Formation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Alessandro Colotti, Angelo Cenedese, Sébastien Briot, Isabelle Fantoni, and Alexandre Goldsztejn

xii

Contents

Braking of a Solid Body Supported by Two Supports on a Horizontal Rough Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Marat Dosaev, Vitaly Samsonov, Liubov Klimina, Boris Lokshin, Shyh-Shin Hwang, and Yury Selyutskiy Workspace Analysis and Torque Optimization on a Schönflies-Motion Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Bruno Belzile and Jorge Angeles Analysis of Running Expansion with Trunk and Pelvic Rotation Assist Suit by Using SLIP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Hongyuan Ren, Takayuki Tanaka, Kotaro Hashimoto, and Akihiko Murai Trajectory Planning Strategy for Multidirectional Wire-Arc Additive Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Markus Schmitz, Carlo Weidemann, Burkhard Corves, and Mathias Hüsing Stiffness Modeling of Planar Robotic Manipulators: Model Reduction and Identifiability of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Shamil Mamedov, Dmitry Popov, Stanislav Mikhel, Alexander Klimchik, and Anatol Pashkevich Forward Kinematic Analysis of a Rotary Hexapod . . . . . . . . . . . . . . . . 486 Alexey Fomin, Anton Antonov, and Victor Glazunov Kinematics of 2-DOF AGVs with Differential Driving Wheels and Caster Wheels Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Mohammadreza Montazerijouybari, Luc Baron, and Sousso Kelouwani Motion Synthesis Using Low-Dimensional Feature Space and Its Application to Inverse Optimal Control . . . . . . . . . . . . . . . . . . . 503 Soya Shimizu, Ko Ayusawa, and Gentiane Venture An Analytical Formulation for the Geometrico-Static Problem of Continuum Planar Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 512 Federico Zaccaria, Sébastien Briot, M. Taha Chikhaoui, Edoardo Idà, and Marco Carricato Static Analysis and Design of Extendable Mechanism Inspired by Origami Structure Based on Non-overconstrained Kinematically Equivalent Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Reiji Ando, Hiroshi Matsuo, Daisuke Matsuura, Yusuke Sugahara, and Yukio Takeda Function Approximation Technique Based Immersion and Invariance Control for an Underactuated Tower Crane System . . . . . . . . . . . . . . . 530 Yang Bai and Mikhail Svinin

Contents

xiii

Dynamic Modeling and Controller Design of a Novel Aerial Grasping Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 Zhongmou Li, Xiaoxiao Song, Vincent Bégoc, Abdelhamid Chriette, and Isabelle Fantoni Dynamically-Feasible Trajectories for a Cable-Suspended Robot Performing Throwing Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Deng Lin, Giovanni Mottola, Marco Carricato, Xiaoling Jiang, and Qinchuan Li Determination of the Geometric Parameters of a Parallel-Serial Rehabilitation Robot Based on Clinical Data . . . . . . . . . . . . . . . . . . . . . 556 Dmitry Malyshev, Santhakumar Mohan, Larisa Rybak, Gagik Rashoyan, and Anna Nozdracheva Optimal Selection of Transmission Ratio and Stiffness for Series-Elastic Actuators with Known Output Load Torque and Motion Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 Guido Bocchieri, Luca Luzi, Nicola Pedrocchi, Vincenzo Parenti Castelli, and Rocco Vertechy Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Active Visualization of Non-destructive Inspection for Metal Using Terahertz Camera and Light Source Shunsuke Yamada(B) , Teppei Tsujita, Masahiro Kurosaki, Tetsuo Tomizawa, Yutaka Sakuma, and Ryosuke Eto National Defense Academy, Yokosuka 239-8686, Japan [email protected]

Abstract. Terahertz waves are electromagnetic waves which are frequency bands ranged from 0.1–10 THz. These waves have high transparency to non-metal, and high reflectivity for the metals. Expected to be application to non-contact exploration sensor for the dangerous goods, bomb, knife, and so on, the development to railway and airport is considered. In this paper we aim to construct a terahertz system that can be mounted on the mobile robots, and to visualize the metal object behind these clothes. Keywords: Terahertz camera · Non-destructive inspection · Active visualization

1 Introduction There have the increasing interests of the detection of hidden weapon in crowds, and the development of the fast screening system for the passenger in airport, station and so on is expected. Over past decade, the terahertz wave has been gathering attention, because it would have high potential of the non-destructive inspection [1]. The terahertz wave is the frequency band ranged from 0.1 THz to 10 THz and the wavelength from 3 mm to 3 µm in Fig. 1. The several points of its advantage are to transmit the non-metallic and non-polar medium, in the other words, ‘see though’, and reflect the metallic and polar medium. Also, this wave band has rarely health risk unlike the baggage inspection using the X ray in airport. The technique for the non-destructive inspection using the terahertz wave is possible to downsize, and to be construct by a few modules such as terahertz camera and light source. The terahertz wave is able to inspect the metallic medium non-destructively. The commercial scanner operating in the terahertz and millimeter wave are most often developed in the security instruments [2, 3]. However, when the security in living space is considered, this security system is better to be dynamic monitoring using the mobile robots than the regular monitoring as fix-point camera. The mobile robot system with the terahertz module in watching the pedestrian is not proposed as shown in Fig. 2. It is necessary to downsize and simplify the terahertz modules and system in order to mount on the mobile robot. Now, there are two type options of the non-destructive © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 1–7, 2021. https://doi.org/10.1007/978-3-030-58380-4_1

2

S. Yamada et al.

Fig. 1. Frequency and wavelength of terahertz waves in the electric radiation band

Hidden weapon Terahertz camera and source

Pedestrian

(a) Mobile robot with terahertz modules

(b) Dynamic monitoring for passengers

Fig. 2. Terahertz radiation of electric radiation band

inspection using terahertz waves. Like as the commercial scanner above, one is the passive visualization [4] which captures the back light of the metallic medium due to the self-radiation emitted from the human body in Fig. 3(a). This option is unsuitable for the mobile robot because it is necessary to use the movable mirror and have high accuracy of the mirror position in this system. But the active visualization is able to build a few modules as the terahertz camera and light source, respectively. This option of active visualization [5] is to capture the terahertz light reflected from the metallic medium and transmitted through the nonpolar medium. It is possible for the active visualization to opaque the thicker cloth by using the strong light source, and to simplify the construction. In this paper, the purposes of this study are to detect the metallic material behind the cloth, and to confirm the field of the view by using the camera and the light source at the first step of the study. We assume the small concealed weapons like the size of 10 × 10 cm2 such as small knife and gun. Terahertz system Hidden weapon

Lens

Mirror

Terahertz camera

Terahertz source

Moving mirror

(a) Passive visualization

(b) Active visualization

Fig. 3. Method of hidden object detection in terahertz band

Active Visualization of Non-destructive Inspection for Metal

3

2 Non-destruction Detection 2.1 Terahertz Camera and Light Source In this study, we employed the terahertz camera (INO, IRXCAM-THz-384 camera, 384 × 288 pixels, 50 Hz, 595 g, 61 × 61 × 65 mm3 ), and light source (INO, THz Illumination @282 GHz, 4 mW, 12.7 kg, 250 × 400 × 440 mm3 ), respectively, as show in Fig. 4(a). Figure 4(b) shows that the non-destructive inspection system for the metal was built on the optical table with the working distance of 1 m from the HDPE (High Density Polyethylene) focal lens. The sampling frequency, which is set at 50 Hz is enough to capture the walking human. In this paper, the experiments with close range were conducted. The sensitive and sampling frequency of this camera were controlled by PC, and the image data was saved at PC. The image data in this paper was conducted in the post process. HDPE focal lens

THz camera & Refractive lens

THz light source

(a) Terahertz imaging system

(b) Arrangement of camera and light source

Fig. 4. Terahertz camera and light source

2.2 Detectable Size of Terahertz Image at 1 m At first, in order to verify the object size and the detectable area in the terahertz image, we captured the front head of the metal arm of the robot manipulator (Universal Robots, UR3), which was movable on the X and Z directions in Fig. 5(a). We set at the distance of 1 m between the HDPE optical lens and the robot arm, because the reflectance brightness from the metal arm has been weak. The reflective area for the terahertz was the surface of the end effector of the robot manipulator with the diameter of 30 mm. As show in Fig. 5(b), the brightness of the object was measured behind the cloth. Fig. 5(c) shows that the object of the robot manipulator moves in the X and Z ranged from −65 to 65 mm and 330 to 420 mm. Between the HDPE lens and this object, there were thin cotton (100% cotton), thick denim (98% cotton, 2% polyurethane). We compared with the reflective brightness of no cloth and cloths. Figure 6 shows a brightness of the inflection image behind (a) no cloth and (b) a denim when the robot manipulator moves to the horizontal direction in the X ranged

4

S. Yamada et al.

from −50 to 50 mm at the stational of Z = 375 mm. In Fig. 6(a), the around center of this image appears on the high value of the inflective brightness in this image, however this value reduces to 75% at (X, Z) = (0.375) if the area would be out of the spot on the specular reflection. As shown in Fig. 6(b), the entire area of this brightness is slightly attenuated, due to the transmittance of the denim.

(a) Terahertz imaging system

(b) Front head of robot manipulator

(c) Movable area Fig. 5. Terahertz camera and light source

(a) No cloth

(b) Denim

Fig. 6. Reflected brightness of terahertz wave at Z = 375 mm

Figure 7 shows a binarized image of Fig. 6 in the case of (a) no cloth and (b) denim. Then, the white region of this binarization sets above the threshold level of 60% for

Active Visualization of Non-destructive Inspection for Metal

5

256 gradations. As shown in Fig. 7(a), we can detect a width of about 30 mm in the Z direction and about 100 mm in the X direction from the reflected brightness. Also, from Fig. 7(b), in the case of the denim, the transmitted light of the terahertz light is slightly attenuated, and the detectable area in the X direction and the Z direction drops to 26% based on that without the cloth.

(a) No cloth

(b) Denim

Fig. 7. Binary images of reflective brightness at Z = 375 mm

Figure 8 shows the accumulated image of the brightness value when the robot manipulator was moved in the range of X = −65 to 65 mm and Z = 330 to 410 mm. The red line indicates the movable area of the robot manipulator. From Fig. 8 (a), the imaging area is not symmetric in terms of Z = 370 mm, because the angle of this camera and light source is a little bit off from the proper location. In Fig. 8 (b), compared to Fig. 8 (a), the terahertz light detection area is reduced, and a rectangular area of about 80mm in the X direction and about 40mm in the Z direction is detected. Therefore, with this measurement system, the size of the flat metal hidden under the denim fabric is about 80 mm wide and 40 mm high.

(a) No cloth

(b) Denim

Fig. 8. Integrated binary images of terahertz wave in Z ranged from 345 to 395 mm

6

S. Yamada et al.

2.3 Reflective Image of Square and Triangular Metallic Pieces Square and triangular metallic pieces with a side length of about 30 mm as shown in Fig. 9 were used as the two shapes of the object. Figure 10 and 11 show the reflective brightness for square and triangular metallic pieces, respectively, without the cloth and with the cotton, and the denim. The rectangular metallic piece in Fig. 10 appears the reflected brightness. Also, the reflective area is slightly smaller than the 30 mm square metal piece, with or without cloth. Also, the shape of this brightness is not clear square.

30mm

(a) Square

30mm

(b) Triangle

Fig. 9. Sample of metric plate

In the triangle case of Fig. 11, the reflective brightness decreases as the thickness of the cloth increases without the cloth, the cotton, and the denim. Also, unlike the actual triangular shape, the shape is close to a square. In this paper, the terahertz light reflected from a metallic object can be detected by a terahertz camera through a cloth assumed the clothing, however it is slightly different from the actual shape. As this cause, there are the several possibilities that the reflection of the terahertz light has been weak, or the distance between the HDPE lens and the terahertz camera is not correct and so on. Also, the resolution of terahertz image has the limitation because of the diffraction-limited system. The image resolution is improved by using the imaging technique for instance, a deep convolutional neural network [6] or the synthesized method of the terahertz and visual images [7]. However, in this paper, the metallic material under the cloths can be detected by the terahertz camera and light source.

Fig. 10. Visualization of square plate

Active Visualization of Non-destructive Inspection for Metal

7

Fig. 11. Visualization of triangle plate

3 Conclusions We constructed the system of the non-destructive inspection for metal using the terahertz camera and light source which enable to mount on the mobile robots. In this study, the terahertz wave reflected from the metal hidden on the cloth was detected by the camera. It is possible to confirm the detectable size and area for the object, when the camera and light source are fixed. In order to improve the image resolution, the post process of the terahertz image is considered in the future. Also, we plan on conducting the terahertz imaging for the long distance, and the identification of the pedestrians carrying the metallic material.

References 1. Federici, J.F., et al.: THz imaging and sensing for security applications-explosives, weapons, and drugs. Semicond. Sci. Technol. 20, S266–S280 (2005) 2. L3. https://sds.leidos.com/advancedimaging/provision-2.htm 3. Smith Detection. https://www.smithsdetection.com/products/eqo/ 4. Kowalski, M.: Real-time concealed object detection and recognition in passive imaging at 250 GHz. Appl. Opt. 58(12), 3134–3140 (2019) 5. Defour, D., et al.: Review of terahertz technology development at INO. J. Infrared Millimeter Terahertz Waves 36, 922–946 (2015) 6. Zhenyu, L., et al.: Terahertz image super-resolution based on a deep convolutional neural network. Appl. Opt. 58(10), 2731–2735 (2019) 7. Liu, Z., et al.: Concealed weapon detection and visualization in a synthesized image. Pattern Anal. Appl. 8, 375–389 (2006)

Model-Based Dynamic Human Tracking and Reconstruction During Dynamic SLAM Huayan Zhang1 , Tianwei Zhang2(B) , and Lei Zhang1 1

Beijing University of Civil Engineering and Architecture, Beijing 100044, China 2 Department of Mechano-Informatics, School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan [email protected]

Abstract. Most of the existed Simultaneous Localization and Mapping solutions cannot work in dynamic environments since the dynamic objects lead to wrong uncertain feature associations. In this paper, we involved a learning-based object classification front end to recognize and remove the dynamic object, and thereby ensure our ego-motion estimator’s robustness in high dynamic environments. The static backgrounds are used for static environment reconstruction, the extracted dynamic human objects are used for human object tracking and reconstruction. Experimental results show that the proposed approach can provide not only accurate environment maps but also well-reconstructed moving humans.

Keywords: Dynamic environment reconstruction

1

· SLAM · Human motion

Introduction

Simultaneous Localization and Mapping (SLAM) is a prerequisite for robots to work in unknown environments. The robots can improve the ability to perceive the environment by using SLAM, thereby assisting them in deciding on tasks. The task of SLAM is to estimate its ego-motion and build an environment map using only sensors on the robot. Most of the existed visual SLAM methods were based on the assumption of the static environment, but the real world is common with dynamic objects which result in a lot of uncertain disturbance to robots [4]. Directly using static SLAM to dynamic scenes will cause bad odometry estimation and mapping results. The reason is that dynamic features hinder the process of SLAM. To deal with dynamic SLAM, the most direct method is to remove the dynamic features and only perform SLAM in the static part. Therefore, the static world assumption can be used to handle dynamic SLAM. In recent years, deep learning has received widespread attention in the application of computer c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 8–15, 2021. https://doi.org/10.1007/978-3-030-58380-4_2

Model-Based Dynamic Human Tracking and Reconstruction

9

vision. The dynamic objects can be effectively detected by applying deep learning methods in a dynamic environment. Most of the dynamic SLAM methods focus on the reconstruction of the static environment and ignore the dynamic objects. However, dynamic objects also help the robot perceive the environment. For example, the robot reconstructs dynamic objects instead of only reconstructing the static environment during SLAM, which may prevent collisions between the robot and dynamic objects such as the human. The human body is the most common object in a dynamic environment. Thus, dynamic human reconstruction is as important as static environment reconstruction. In this paper, we propose a method for reconstructing the human body and tracking human motion during dynamic SLAM in the meantime. The dynamic SLAM is implemented based on ORB-SLAM2 [9], and the feature selection module of ORB-SLAM2 is changed to select only the ORB (Oriented FAST and Rotated BRIEF) features in the static background to maintain the robustness of ego-motion estimator. The dynamic objects are segmented by using a learning-based method, and the extracted human bodies are used for tracking and reconstruction. Therefore, the proposed method is capable of simultaneously outputting a static map of the environment and the 3D mesh models of each dynamic human body. Our method was tested on the public datasets, and the experiments show that the proposed method can provide not only accurate environment maps but also well-reconstructed moving humans.

2

Related Works

At present, many researchers have focused on the solutions of dynamic SLAM. Most of existed dynamic SLAM methods were based on feature-based or direct SLAM. In this paper, we divide the related work into two types: dynamic segmentation based and object classification based. Among the object classification based SLAM works, some researches used semantic labels to remove dynamic objects as exceptions with the help of deep learning. Zhang et al. proposed PoseFusion in [17] for segmenting dynamic human body and static background reconstruction. Bescos et al. [1] proposed DynaSLAM that used Mask-RCNN [6] for dynamic object detection. On the dynamic segmentation side, basing on an RGB-D static SLAM method ElasticFusion (EF) [14], some studies have attempted to find the dynamic point clouds as exceptions in EF’s dense RGB-D fusion processing. Scona et al. [12] achieved the reconstruction of static background and improved accuracy and robustness in dynamic scenes. Zhang et al. proposed FlowFusion in [18], which applied optical flow residuals to dynamic object segmentation and reconstruct the static background according to the static keyframe recording. The above SLAM methods have obtained good experimental results in dynamic environments. The general solution for dynamic object reconstruction in dynamic environments is to explicitly model the scene non-rigidity. The reconstruction of general non-rigidly deforming objects/scenes based on realtime depth sensor data has a long tradition. One of the traditional methods

10

H. Zhang et al.

Fig. 1. System flowchart: First, the RGB image is input to YOLACT [2] for instance segmentation, and it is divided into the human body and static background. Then, the OpenPose [3] and HMR [8] are jointly used to generate 3D mesh for each human body. Meanwhile, only the static ORB feature are selected for simultaneous localization and mapping in ORB-SLAM2 [9]. Finally, human reconstruction and the static map are obtained at the same time.

use the pre-scanned template and transforming the template to live frames from RGB-D or stereo camera data. Based on the volumetric approach, DynamicFusion [10] is the first template-free on-the-fly non-rigid 3D reconstruction system. Zollh¨ ofer et al. [7] proposed VolumeDeform, which employs a flip-flop data-parallel optimization schema that tackles the non-rigid registration at realtime rates. Besides, they improve tracking robustness using global sparse correspondence from hand-engineered feature matching. DoubleFusion [15] achieve good body tracking results by leveraging the human body model. SurfelWarp [5] reduced the memory and computation cost by employs Surfel rather than a TSDF volume as the 3D representation. R¨ unz et al. proposed Co-Fusion [11] for simultaneously tracking and reconstructing the object’s 3D shape and environment. Similar to Co-Fusion, our method output reconstructed static maps and 3D models of dynamic objects while tracking dynamic objects during SLAM. The difference is that we implement human tracking in feature-based SLAM and reconstruct only dynamic human bodies using the model-based method.

3

System Overview and Method

In this study, the proposed method can simultaneously track the dynamic human motion and reconstruct the 3D mesh model of human bodies during the process of SLAM in a dynamic environment. Figure 1 shows the system framework of the proposed method. To track the human motion, we use a learning-based method to maintain the robustness of SLAM by masking the dynamic ORB features in the ORB-SLAM2 [9] and extract all the human bodies from a dynamic environment. On this basis, we added a new thread to handle dynamic human

Model-Based Dynamic Human Tracking and Reconstruction

11

motion reconstruction based on HMR [8], and finally output static map and all the human reconstruction. 3.1

Dynamic Human Body Extraction and Static Background Segmentation

For each input RGB image, we use YOLACT [2] to detect and segment objects in the scene. YOLACT is a one-stage instance segmentation method, which can detect and segment 80 types of objects in the image. It splits the task of instance segmentation into two parts and runs them at the same time. The first task generates prototype masks and the second task is object detection based on anchor. The results of these two tasks are combined in a linear combination to generate the final mask. Once we get the final mask, we can segment the objects in the image. In our system, we extract the human body image for human reconstruction and remove the dynamic objects to get the static background according to the object’s mobility. When the input image I is given, the mask is generated by YOLACT and we formate the output data O: O = [c × l]I

(1)

In which c is the classes of the object detected by YOLACT. The parameter l is the location, id, and confidence of the detected object. For each l, it contains four elements: u, v, id, c. The u and v are the 2D pixel coordinate value of the object in the image plane, the id is used to distinguish the different objects in the same class and the c ∈ (0, 1] is the confidence. We only store detected objects with confidence c > 0.5 to save memory space. The pixel of each human can be extracted according to O in the RGB image, and we use it as input for the human reconstruction module. For static backgrounds, we remove moving objects such as humans, chairs, etc. from the environment. 3.2

Dynamic Human Motion Tracking and Reconstruction

To track the human motion, we use the static background as a reference for ORB-SLAM2 [9] to select the static ORB features and reconstruct the static map. For each segmented human image, OpenPose [3] and HMR [8] are jointly used to reconstruct the 3D mesh model of the human body. HMR is a model-based method for reconstructing the 3D human body through the 2D image. It can directly inference 3D meshes of people in image features instead of the skeleton, which retains more characteristics of the human body. The input image uses the Convolutional Neural Network to extract convolutional features and then uses a network structure similar to Generative Adversarial Networks (GAN) to generate a human body model. But HMR only supports the reconstruction of a single human body in the image, and the background environment has a great impact on HMR. On this account, we remove

12

H. Zhang et al.

objects other than the human body from the environment and save each human body separately. When we get segmented images only having one human, we use them as the input of HMR. Each segmented images will be inputted into HMR to get the 3D mesh of the human body. However, directly using HMR may get human reconstruction results with the wrong pose. To ensure the stability of the reconstruction results, OpenPose is used to estimate the 2D human pose. By using both HMR and OpenPose to reconstruct the human body, the stable 3D mesh of the human body can be reconstructed. After the above steps, we not only get a map of the static background, but also a reconstructed human model. The human motion can also be tracked at the same time.

4

Experimental Results and Discussions

To evaluate the rigid mapping results, we compared the proposed method to three state-of-the-art dynamic environment reconstruction methods: CoFusion(CF) [11], StaticFusion(SF) [12] and FlowFusion(FF) [18]. The Absolute Trajectory Errors (ATE)[13] was used as the evaluation metric for evaluating accuracy. It points out the error between the estimated trajectory and the ground truth trajectory. Table 1 shows the ATE RMSE on TUM [13] and HRPSlam [16] RGB-D datasets. The root mean square error (RMSE) indicates the stability of ATE. The first two fr1 sequences are static environments, all of these methods got similar results. The lower four rows are dynamic sequences. As our method applied advanced dynamic object detection and removal technique, VO errors are much smaller than CF and competitive to SF and FF. The proposed method results in the smallest 2.20 cm ATE in fr3/walking xyzsequence. CF works well in

(a) ATE of ours

(b) RPE of ours

Fig. 2. ATE and RPE estimated by our method in the high dynamic sequence fr3/walking xyz. Our method achieves 2.20 cm for ATE RSME and 3.43 cm/s for RPE RSME.

Model-Based Dynamic Human Tracking and Reconstruction

13

Table 1. Absolute Trajectory Error (ATE) RMSE (m) Sequence

CF

SF

fr1/xyz fr1/desk2

0.014 0.017 0.020 0.015 0.17 0.051 0.034 0.040

fr3/walk xyz fr3/sitting xyz HRPSlam2.1 HRPSlam2.4

0.71 0.027 0.91 0.63

0.21 0.043 0.25 0.44

FF

0.12 0.034 0.23 0.49

Ours

0.022 0.013 0.22 0.45

(a) First Human Reconstruction

(b) Second Human Reconstruction Fig. 3. 3D reconstruction of the detected human objects in TUM fr3/walking xyz sequence. Two people were reconstructed separately.

14

H. Zhang et al.

known environments. There are moving objects in this sequence from the beginning, which result in CF’s VO failure. Figure 2 plots the result of ATE and RPE (Relative Pose Error) in fr3/walking xyz sequence. In the plotted ATE image, the blue line represents the estimated trajectory, the black line represents the ground truth trajectory, and the red line is the difference. The plotted image indicates that our method maintains the robustness in this sequence. The reconstructed moving human bodies are shown in Fig. 3. In the fr3/walking xyz sequence, the proposed method successfully reconstructs two people moving at high speed. Because of the lack of the human reconstruction baseline, we only show the reconstruction results without evaluation. The result of human reconstruction indicates that our method can reconstruct the 3D mesh model of each human body in a dynamic environment.

5

Conclusions

In this paper, we proposed a non-rigid human object extracting, tracking and reconstruction method in a dynamic environment based on object classification and dynamic SLAM. We segment dynamic objects based on deep learning and extract a static background for SLAM. Our method can simultaneously output the reconstruction of the human using model-based method and a static map during dynamic SLAM. Experimental results show that the proposed approach can provide not only accurate environment maps but also well-reconstructed moving humans. Acknowledgement. This work is supported by the National Natural Science Foundation of China (Grant No. 61473027), Beijing Key Laboratory of Robot Bionics and Function Research (Grant No. BZ0337) and Beijing Advanced Innovation Center for Intelligent Robots and Systems.

References 1. Bescos, B., F´ acil, J.M., Civera, J., Neira, J.: Dynaslam: tracking, mapping, and inpainting in dynamic scenes. IEEE Robot. Autom. Lett. 3(4), 4076–4083 (2018) 2. Bolya, D., Zhou, C., Xiao, F., Lee, Y.J.: Yolact: real-time instance segmentation. In: Proceedings of the IEEE International Conference on Computer Vision (2019) 3. Cao, Z., Hidalgo, G., Simon, T., Wei, S.E., Sheikh, Y.: Openpose: realtime multi-person 2D pose estimation using part affinity fields. arXiv preprint arXiv:1812.08008 (2018) 4. Chen, X., Yu, Z., Zhang, W., Zheng, Y., Huang, Q., Ming, A.: Bioinspired control of walking with toe-off, heel-strike, and disturbance rejection for a biped robot. IEEE Trans. Industr. Electron. 64(10), 7962–7971 (2017) 5. Gao, W., Tedrake, R.: Surfelwarp: efficient non-volumetric single view dynamic reconstruction. arXiv preprint arXiv:1904.13073 (2019) 6. He, K., Gkioxari, G., Doll´ ar, P., Girshick, R.: Mask r-cnn. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 2961–2969 (2017)

Model-Based Dynamic Human Tracking and Reconstruction

15

7. Innmann, M., Zollh¨ ofer, M., Nießner, M., Theobalt, C., Stamminger, M.: Volumedeform: real-time volumetric non-rigid reconstruction. In: ECCV, pp. 362–379. Springer (2016) 8. Kanazawa, A., Black, M.J., Jacobs, D.W., Malik, J.: End-to-end recovery of human shape and pose. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7122–7131 (2018) 9. Mur-Artal, R., Tard´ os, J.D.: ORB–SLAM2: an open-source slam system for monocular, stereo and RGB-D cameras. IEEE Trans. Rob. 33(5), 1255–1262 (2017). https://doi.org/10.1109/TRO.2017.2705103 10. Newcombe, R.A., Fox, D., Seitz, S.M.: Dynamicfusion: reconstruction and tracking of non-rigid scenes in real-time. In: CVPR, pp. 343–352 (2015) 11. R¨ unz, M., Agapito, L.: Co-fusion: real-time segmentation, tracking and fusion of multiple objects. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 4471–4478. IEEE (2017) 12. Scona, R., Jaimez, M., Petillot, Y.R., Fallon, M., Cremers, D.: Staticfusion: background reconstruction for dense RGD-B SLAM in dynamic environments. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 1– 9 (2018) 13. Sturm, J., Engelhard, N., Endres, F., Burgard, W., Cremers, D.: A benchmark for the evaluation of RGD-B SLAM systems. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 573–580 (2012) 14. Whelan, T., Salas-Moreno, R.F., Glocker, B., Davison, A.J., Leutenegger, S.: Elasticfusion: real-time dense SLAM and light source estimation. Int. J. Robot. Res. 35(14), 1697–1716 (2016) 15. Yu, T., Zheng, Z., Guo, K., Zhao, J., Dai, Q., Li, H., Pons-Moll, G., Liu, Y.: Doublefusion: real-time capture of human performances with inner body shapes from a single depth sensor. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7287–7296 (2018) 16. Zhang, T., Nakamura, Y.: Hrpslam: a benchmark for RGB-D dynamic SLAM and humanoid vision. In: Proceedings of the IEEE International Conference on Robotic Computing, pp. 110–116 (2019) 17. Zhang, T., Nakamura, Y.: Posefusion: dense RGB-D SLAM in dynamic human environments. In: Xiao, J., Kr¨ oger, T., Khatib, O. (eds.) Proceedings of the 2018 International Symposium on Experimental Robotics, pp. 772–780. Springer International Publishing, Cham (2020) 18. Zhang, T., Zhang, H., Li, Y., Nakamura, Y., Zhang, L.: FlowFusion: dynamic dense RGB-D SLAM based on optical flow. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 7322–7328 (2020)

Development of Robust Ridge Detection Method and Control System for Autonomous Navigation of Mobile Robot in Agricultural Farm Shunsuke Fujita1 , Takanori Emaru2(B) , Ankit A. Ravankar2(B) , and Yukinori Kobayashi1,2 1 2

Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan Research Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan {emaru,ankit}@eng.hokudai.ac.jp

Abstract. In this paper, we propose a robust ridge detection method for autonomous navigation of a mobile robot in agricultural fields using an inexpensive RGB-D camera system. In this system, the ridge centerline is estimated using the depth images from RGB-D sensor by using our proposed methods. We propose an autonomous driving system utilizing digital optimal control regulator for high-precision motion and robust recognition of ridge features. From the experimental evaluation, an accuracy level of ±1.5 cm robot traversal along the farm ridge was recorded. Keywords: Autonomous mobile robot vision · Ridge detection

1

· Agriculture robot · Computer

Introduction

In recent years, there has been a serious decline in the population of farmers in developed countries particularly in Japan due to falling birthrate and aging population. At the same time in order to keep up with the international market competitiveness of Japanese agricultural products, it is necessary to reduce the cost of agricultural work. Against the background of these problems and issues, robotization of agriculture task is proposed by the government and researchers to solve this problem. Most existing methods used for autonomous motion control such as robot tractors include RTK-GNSS such as AGI3 [1,2] or high-accuracy satellite positioning system that uses signals from the quasi-zenith satellite, Michibiki that was recently introduced in Japan [3]. These technologies can move within the field with an accuracy of ±5 cm [3], but the receivers for using them are very costly. In this research, it is considered that using a low-cost sensor is realistic to increase the relative positional accuracy with the crop to be worked because absolute accuracy is not important for weeding and harvesting work in the field. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 16–23, 2021. https://doi.org/10.1007/978-3-030-58380-4_3

Ridge Detection Method and Control System for Robot in Agricultural Farm

17

We propose a method of autonomously driving in a field with an accuracy of ±1.5 cm by recognizing a ridge using only one inexpensive active IR stereo RGB-D sensor for less than two hundred US dollars. The novelty is the method of autonomous driving by installing the RGB-D sensor vertically downward.

2

Background

Some robot systems that move in the farm field using RGB cameras without using RTK-GNSS and high-accuracy satellite positioning system have been proposed in the past. Andrew et al. proposed a method of estimating the position of crop rows by installing multiple RGB cameras horizontally and photographing multiple crop rows at the same time and processing the images [4]. In NARO’s automatic steering system that assists tractors in straight line work, one RGB camera was used to recognize work marks and marker marks in the previous process [5]. However the accuracy of these methods was not within ±5 cm. In our method, the RGB-D sensor was installed downward to recognize one ridge, and thereby improving the relative position accuracy with the ridge and the crops, so it achieved an accuracy within ±5 cm.

3

System Architecture

In this research, we made a two-wheeled differential robot that can move across ridges to realize autonomous movement on a farm. The dimensions are as shown in Fig. 1. An RGB-D sensor is installed facing down in front of the robot, to collect data of the ridges as the robot moves along it. The robot has two passive wheels for better stability and motion in the agriculture field environment. We used RealSense D435, an active IR stereo RGB-D sensor made by Intel Corporation (hereafter referred to as RGB-D sensor) as the main sensor. To obtain ground truth for evaluating experimental results, we also used a 2D LiDAR, UST-20LX made by Hokuyo Electric Co., Ltd.

Fig. 1. Robot dimensions

Fig. 2. Positional relationship between RGB-D sensor and 2D LiDAR

18

S. Fujita et al.

4

Method

We built the software using opensource middleware called Robot Operating System (ROS) and used ROS Kinetic on Ubuntu 16.04 operating system. The system configuration is as shown in Fig. 3, and ridge center estimator, robot controller, and wheel controller are the ROS nodes developed for this research.

Fig. 3. System configuration

Fig. 4. Ridge estimation from RGB-D image

Fig. 5. Depth image of a ridge

Fig. 6. Schematic diagram of ridge region estimation

4.1

Ridge Center Estimator

As shown in Fig. 3, after receiving the data of the RGB image and the depth image from the RGB-D sensor, this node estimates the center line of the ridge and distributes the center line information based on the line. The estimation is conducted in three steps by three algorithms described in detail below. The first step is extracting the ridge region, second step is calculating the average center line using the region, and last step is obtaining the approximate center line by the least squares method based on the green line in Fig. 4. The red region is the ridge region estimated using the depth information from the sensor.

Ridge Detection Method and Control System for Robot in Agricultural Farm

19

Algorithm for Ridge Region Estimation: Figure 5 shows the depth image of the ridge corresponding to the RGB data of Fig. 4. The estimation of the ridge region is performed by the threshold value calculated by Eq. (1). In this equation, max depth is the maximum distance in one depth image, and min depth is the minimum distance. 0.9 is the weight parameter determined by trial and error. As a specific estimation procedure, as shown in Fig. 6, a point whose distance value is smaller than the threshold is considered as a ridge region. This test is performed for all the pixels. threshold =

max depth + min depth × 0.9 2

(1)

Algorithm for Obtaining the Average Center Line of the Ridge Region: The average center line of the ridge is calculated based on the ridge region obtained in the previous step. As shown in Fig. 6, it adds the coordinate values from the left edge of the pixel, which is in the ridge region, and takes the average position. It performs this operation from the top to the bottom of the depth image. Algorithm for Obtaining the Approximate Center Line of Ridge: A straight line approximation of the center line of the ridge is performed based on the average center line of the ridge obtained in the previous step. The coordinate is defined as shown in Fig. 7. For the linear approximation, the least squares method was used, and an approximate straight line was expressed as below. y = ra x +

cd 2

(2)

cd is defined as the distance from the center of the sensor image to the center of gravity of the approximate center line of the ridge and ra is defined as the gradient of the approximate center line as shown in Fig. 8. These are published to Robot Controller node as ridge position information viewed from the robot. 4.2

Robot Controller

This node receives the ridge position information from Ridge Center Estimator and distributes x˙l , the target translation speed of the robot, and z˙a , the target rotational speed around the center axis of that, to Wheel Controller based on it. In this research, we designed the digital optimal regulator control system with two inputs and one output. We set cd and ra , which were obtained as input and z˙a , the rotational speed, as output whereas x˙l is a constant. Control Using Digital Optimal Regulator: The discretized state equation of the robot can be expressed as Eqs. (3) to (5). x[k + 1] = Ad x[k] + Bd u[k]

(3)

20

S. Fujita et al.

Fig. 8. Definition of cd , ra

Fig. 7. Coordinate definition of depth image



y[k] x[k] = θ[k]



  u[k] = z˙a [k]

(4) (5)

For this state equation, the optimal input uopt that minimizes the discretized evaluation function, Eq. (6), is obtained by solving the positive definite symmetry of the Riccati equation, Eq. (7), by iterative calculation and finding the gain of Eq. (8). ∞ 1 T [x [k]Qx[k] + uT [k]Hu[k]] (6) J= 2 k=1

P =Q+

ATd P Ad

− ATd P Bd [H + BdT P Bd ]−1 BdT P Ad

K = [H +

BdT P Bd ]−1 BdT P Ad

uopt [k] = −Kx[k]

(7) (8) (9)

Robot Controller node distributes z˙a [k] = uopt [k], which is the target rotation speed of the robot, to Wheel Controller based on the above control system. 4.3

Wheel Controller

This node operates on the microcomputer, Arduino Mega. It receives x˙l and z˙a sent from Robot Controller node, converts the speed and performs PD control of wheels.

5

Modeling

In this section, we explain the exact modeling of the robot we have created.

Ridge Detection Method and Control System for Robot in Agricultural Farm

5.1

21

Robot

Figure 9 shows a schematic diagram of the robot. The sensor is fixed on an extended arm called the sensor arm. The length of that arm is defined as R, and the tread width is defined as 2d. 5.2

Coordinate System

The coordinate system for the robot is determined as shown in Fig. 10, with X and Y as global coordinates. The X-axis is set to overlap the center of the ridge and Y = 0. The robot coordinate are indicated as (xb , yb ) and the sensor coordinate as (xs , ys ) with respect to the global frame. The sensor frame is R distance from the robot frame and perpendicular to the front wheel of the robot. θ indicates the inclination of the sensor coordinate system to the global coordinate system. The counterclockwise direction is the positive direction.

Fig. 9. Differential two-wheeled model

Fig. 10. Coordinate frame

Fig. 11. Travel speed and global coordinate system

Fig. 12. Movement during turning

22

5.3

S. Fujita et al.

Modeling

At first, we modeled a two-wheeled differential robot in a continuous system.  y˙s Forward speed is x˙l , and rotation speed is z˙a . The state equation of is θ˙ obtained as follows.        y˙s ys −R 0 x˙l (10) + z˙ = 0 0 θ 1 a θ˙ We performed Euler transformation for the state Eq. (10). We set sampling time as T [s]. The discretized state equation of the differential two-wheeled robot is as follows.        1 x˙l T ys [k] −RT ys [k + 1] (11) = + z˙ θ[k + 1] 0 1 θ[k] T a

6 6.1

Traveling Experiment in Farm Field Experiment Contents

In an outdoor farm experiment, it was verified whether the robot could run along the ridge of the actual field using control by the digital optimal regulator. The robot was programmed to run over the ridge such that the sensor arm is directly over the ridge. 6.2

Results

Figure 14 shows the results of the traveling experiment. The measurement result of the RGB-D sensor was obtained with an accuracy of ±20 pix, ±3.1 cm, and a ridge angle of ±0.152 rad. The results of 2D LiDAR measurements confirmed that the robot travelled with an accuracy of ±1.5 cm. Note that the two sensors are mounted in different locations, and the results of 2D LiDAR represent the driving accuracy.

7

Summary

In this research, we constructed a control system with an active IR stereo RGBD sensor installed facing down in front of a robot for the purpose of recognising a ridge and running along it on farm fields. An active IR stereo RGB-D sensor was effective for outdoor ridge recognition, and robust ridge recognition was realized by our proposed method. In addition, a two-input, one-output control system using a digital optimal regulator was used to drive the center of the ridge with high accuracy with a differential two-wheeled robot. From our tests, the proposed method using only inexpensive RGB-D sensors as external sensors in a farm environment with ridges can be an option for highly accurate autonomous movement that is indispensable for automation of agricultural work. In future work, it is necessary to develop more experiments to make a deeper quantitative evaluation of the method.

Ridge Detection Method and Control System for Robot in Agricultural Farm

23

Fig. 13. Traveling experi- Fig. 14. Distance from center(RGBment D)[cm] (left), angle of ridge(RGB-D)[rad] (center) and distance from center(2D LiDAR)[cm] (right)

References 1. Noguchi, N., Takai, R., Yang, L.: Development of crawler-type robot tractor based on GNSS and IMU. Eng. Agric. Environ. Food 7(4), 143–147 (2014) 2. Underwood, J.P., Calleija, M., Taylor, Z., Hung, C., Nieto, J.M.G., Fitch, R., Sukkarieh, S.: Real-time target detection and steerable spray for vegetable crops (2015) 3. Noguchi, N.: Current status and future prospects for vehicle robotics and agriculture. J. Jpn. Soc. Precis. Eng. 81(1), 2015 (2015) 4. English, A., Ross, P., Ball, D., Corke, P.: Vision based guidance for robot navigation in agriculture. In: 2014 IEEE International Conference on Robotics and Automation (ICRA) (2014) 5. Hanawa, K.: Automatic steering system to assist tractor straight line work. Sugar & Starch Information, May 2017 6. Ravankar, A., Ravankar, A., Emaru, T., Kobayashi, Y.: Low-cost mobile platform for ROS. In: The Proceedings of JSME Annual Conference on Robotics and Mechatronics (Robomec) 2017, pp. 1P1–G09. The Japan Society of Mechanical Engineers (2017) 7. Ravankar, A.A., Ravankar, A., Kobayashi, Y., Emaru, T.: Autonomous navigation of ground robot for vineyard monitoring. In: The Proceedings of JSME Annual Conference on Robotics and Mechatronics (Robomec) 2019, pp. 1A1–E05. The Japan Society of Mechanical Engineers (2019)

Continuous Jumping Control Based on Virtual Model Control for a One-Leg Robot Platform Libo Meng1,2(B) , Marco Ceccarelli2,3 , Zhangguo Yu1,2 , Xuechao Chen1,2 , Gao Huang1,2 , and Qiang Huang1,2 1

2 3

Beijing Institute of Technology, Beijing 100081, China [email protected] Beijing Advanced Innovation Center for Intelligent Robots and Systems, Beijing Institute of Technology, Beijing 100081, China LARM2: Laboratory of Robot Mechatronics, Deptartment of Industrial Engineering, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy

Abstract. In this paper, a jumping control method is proposed as based on Virtual Model Control (VMC). The virtual force and torque acting on the trunk are used to control the jumping height, speed, and posture. VMC is used to calculate the driving torque of joints to exert the desired virtual force and torque. Each jumping cycle is divided into three states: flying in the air, landing and balance control, and thrusting. Then, a finite state machine (FSM) is adopted to switch between each jumping states. The proposed continuous jumping method is verified through dynamic simulations whose results are discussed also to characterize the jumping performance. Keywords: Continuous jumping control · Infinite state machine.

1

· Virtual Model Control · Torque

Introduction

Legged robots, especially the bipeds, are always at the frontier of robotic fields. The legged movement features these robots more adaptable than the wheeled robots in complex environments. In recent years, great progress and impressive results have been made in stable walking and dexterity manipulation [1–6]. However, the bipedal robots still cannot reach the movement capabilities to their human counterparts, especially for the high dynamic running and jumping. Jumping is the basic element of running since the running can be combined of multiple jumping cycles [7]. Raibert’s team is one of the groups that first studied the jumping for legged robots. The controller of most of the existing quadruped robots are based the basic idea from Raibert’s method [8]. Thanks to the hydraulic drivers, the robot Atlas from Boston Dynamics showed amazing jumping and running ability. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 24–33, 2021. https://doi.org/10.1007/978-3-030-58380-4_4

Continuous Jumping Control

25

Unfortunately, their technical details were not disclosed, and some researchers argued that the motion of Atlas is not natural enough [9]. For the electric driven biped robots, Urata et al. used high speed and torque motor to increase the joints power of HRP3L-JSK robot and realized vertical jumping [10]. The lack of shock-absorbing components prevents the HPR3L-JSK from landing on the ground. Jiang et al. introduced the ZMP as a constraint to control the robot jumping, so that they realize the simulation of vertical jumping and stable landing [11]. Elastic mechanism is an efficient way to enhance the thrust force for jumping motion, it can store the energy during landing then release the energy when the robot leaves the ground. Niiyama et al. used springs to design a robot leg which can jump two times of its body height [12]. Hondo et al. add a spring mechanism to a NAO robot to provide the jumping force [13]. Haldane et al. adopted Series-Elastic-Actuator to the Salto robot which can mimic the vertical jumping of night monkey [14]. For the above mentioned researches, good results about robot jumping is one side of the coin, instability is often the other side. None of these robots can walk or run very well. In this paper, a control method for continuous jumping for a one-leg robot platform is introduced. This one-leg robot platform is driven by electric motors and low reducer ratio. Thus, it has good characteristic of backdrivability. No sensor information except the joint angle is needed for the controller. The virtual force and torque acting on the trunk are distributed to the output torque of each joint based on Virtual Model Control. The stability, jumping height, and jumping speed can be controlled by changing the virtual force and torque on the trunk. The proposed control is verified through dynamic simulations of a vertical jumping, and a forward and backward jumping.

2

Dynamic Model for One-Leg Jumping Platform Based on VMC

In this section, the dynamic model of the one-leg robot platform is in Fig. 1 introduced. A simplified three-link model is used to calculate the virtual force and torque on the center of mass of the trunk. The virtual force and the torque are then distributed into the control torque of each joint of the robot. As is shown in Fig. 1, the robot on the right hand is the one-leg robot platform. It has 3 DoFs, the ankle, the knee, and the hip. Each of these three joints is driven by electric motors that has large output torque and low reducer ratio. Thus, the joints have good backdrivability which enable the complaint landing when compared with the robot using large reducer ratio. The simplified four-link model in Fig. 1 (b) is used to described the robot dynamic parameters. L1 , L2 , L3 are the length of the calf, the thigh, and the trunk, respectively. M1 , M2 , M3 are the mass of the calf, the thigh, and the trunk, respectively. Supposed that during the jumping, the foot of the robot is fully attached on the ground, θ is the posture of the trunk relative to the world coordinate.

26

L. Meng et al.

M3

M2

M1

(a)

(b)

Fig. 1. A dynamic model for the one-leg jumping platform: a) a kinematic scheme; b) a CAD design Fz M

Fx

Fig. 2. Virtual force distributed based on VMC

According to [8], a jumping cycle has three parts: hopping, forward speed, and body attitude. This controller is efficient for the robot with point feet and that the leg has much less mass than the body. Because that the effect of fast leg movement can be ignored to the jumping. Moreover, the balance control is not necessary during the standing period. These two drawbacks should be taken into consideration when the one-leg robot platform to jump. Hence, we introduce the VMC to control the trunk to achieve the dynamic jumping with the one-leg robot platform. As is shown in the Fig. 2, We assume that the force in vertical direction Fz , the force in horizontal direction Fx , and the torque in sagittal plane M are act

Continuous Jumping Control

27

at the COM of the trunk. x, z is the position of the COM of the trunk given by ⎧ ⎪ ⎪ x = L1 sin(q1 ) + L2 sin(q2 ) + l3 sin(q3 ) ⎪ ⎪ ⎨ z = L1 cos(q1 ) + L2 cos(q2 ) + l3 cos(q3 ) (1) ⎪ ⎪ ⎪ ⎪ ⎩ θ = q1 + q 2 + q 3 Then, the Jacobian from the joint space to the Cartesian space can be calculated by deriving both sides of Eq. 1 as: ˙ T = J [q˙1 , q˙2 , q˙3 ]T [x, ˙ z, ˙ θ]

(2)

The virtual force and torque acting on the COM of the trunk can be distributed to each joint of the leg as: T

[τ1 , τ2 , τ3 ] = J T [fx , fz , M ]

T

(3)

where τ1 , τ2 , τ3 is the output torque of the ankle joint, the knee joint, and the hip joint, respectively. τ1 , τ2 , τ3 can be calculated through the desired position [xd , zd ] and the desired posture τd of the COM of the trunk in the follow: M = kpM · (θ − θd ) + kvM · θ˙

(4)

fx = kpx · (x − xd ) + kvx · x˙

(5)

fz = kpz · (z − zd ) + kvz · z˙

(6)

Jumping

Flying

Landing

Fig. 3. An example of human jumping analysis

28

3

L. Meng et al.

Finite State Machine for Continuous Jumping

In the Fig. 3, a typical jump of human includes three stages namely: jumping, flying, and landing, where the posture leaving the ground is almost the same to the posture landing on the ground. Thus, a jumping cycle of a robot also consists of three stages namely: thrusting, flying in the air, and landing and balance control as represented in Fig. 4. Continuous jumping includes many above mentioned jumping cycles. Thus, an event-based finite state machine (like in [15]) is introduced to switch between the three stages. Since the one-leg jumping platform has no sensors except the encoders in each joint, we estimate the triggers just according to the displacement of the trunk relative to the ankle which can be calculated from the joints angles.

State 1：PD Control in the air

State 2：Landing and Balance Control

State 3：Jumping Control

Fig. 4. Three states of a jumping cycle

As inspired by the human jumping, a PD controller is used to turn the posture of the robot in the air back to what the robot is at the very beginning of the jumping: ⎧ ⎪ ⎪ τ1 = kp air1 · (q1 − q1 init ) + kv · q˙1 ⎪ ⎪ ⎨ τ2 = kp air2 · (q2 − q2 init ) + kv · q˙2 (7) ⎪ ⎪ ⎪ ⎪ ⎩ τ3 = kp air3 · (q3 − q3 init ) + kv · q˙3

Continuous Jumping Control

29

When the robot is at landing and standing configuration on the ground, Eqs. 4–6 can be used to control the robot. In order to keep the robot dynamically balanced, we set zd = zinit , xd = 0, θd = 0. Thus the robot can keep the center of pressure within the support polygon and go back to the initial configuration of jumping. Feed-forward compensation (δτ1 , δτ2 , δτ3 ) including gravity and Coriolis force is used to reduce the PD parameters and make the motion smooth: ⎧ ⎪ ⎪ τ1 = τ1 + δτ1 ⎪ ⎪ ⎨ τ2 = τ2 + δτ2 (8) ⎪ ⎪ ⎪ ⎪ ⎩ τ3 = τ3 + δτ3 A spring-dumping model is used to generate the thrusting for the robot to leave the ground with a model as: fz = kp jump · (q2 − q2 jump ) + kv

jump

· q˙2

(9)

where the q2 jump is the desired angle of the knee joint when the robot is at the leaving configuration the ground.

1

2

3

4

Fig. 5. Screen-shots of a continuous vertical jumping

30

L. Meng et al.

An event-based jumping cycle control is used to switch among the above mentioned three jumping states. Algorithm 1 provides the details of the controller as in the following instruction: Algorithm 1 Jumping cycle control. Input: The ankle joint angle q1 , the knee joint angle q2 , and the hip joint angle q3 ; Output: The output torque of ankle joint τ1 , the output torque of knee joint τ2 , The output torque of hip joint τ3 ,; 1: Set the jump flag Jump = 0, state flag State = 2; 2: Calculate the distance between the ankle and trunk D = sqrt(2cos(q2 )L1 L2 + L21 + L22 ); 3: if D > Dlanding &&state == 2 then 4: state = 2 5: else 6: state=1 7: end if 8: switch (state) case 1: 9: M = kpM · (θ − θd ) + kvM · θ˙ 10: fx = kpx · (x − xd ) + kvx · x˙ 11: if D > Djumping &&jump == 0 then 12: fz = kpz · (z − zd ) + kvz · z˙ 13: else 14: if D < Dleaving then 15: fz = kp jump · (q2 − q2 jump ) + kv jump · q˙2 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26:

jump = 1 else jump = 0, state = 2 end if end if T T [τ1 , τ2 , τ3 ] = J T [fx , fz , M ] case 2: τ1 = kp air1 · (q1 − q1 init ) + kv · q˙1 τ2 = kp air2 · (q2 − q2 init ) + kv · q˙2 τ3 = kp air3 · (q3 − q3 init ) + kv · q˙3 end switch return τ1 , τ2 , τ3 ;

4

Simulations

A robot model that has the same parameters with the real robot is built in the simulation environment. The mass of trunk, thigh, and calf are 13 kg, 6 kg, and 4 kg, respectively. The maximum output torque of the hip joint, the knee joint,

Continuous Jumping Control

31

and the ankle joint is 300 Nm, 300 Nm, and 200 Nm, respectively. Two simulations are reported to verify the proposed method namely: a vertical jumping, and a continuous jumping.

1

2

3

4

5

6

7

8

9

Fig. 6. Screen-shots of continuous forward and backward jumping

foot trunk

Height(Meters)

0.8 0.6 0.4 0.2 0 0

1

2

3

4

Time(Seconds) Fig. 7. Computed height of trunk and foot during robot jumping

32

L. Meng et al.

Figure 5 shows the screen-shots of one cycle of a jumping. The first picture in Fig. 5 is the beginning of a jumping cycle. 1 in Fig. 5 are the state of flying in the air, the robot keeps the original posture and detects the landing trigger. When the distance between the ankle and the trunk D is smaller than Dlanding , the robot starts the landing and balance control (2 in Fig. 5). 3 in Fig. 5 are within the jumping cycle. At last, the robot starts the next jumping cycle as shown in 4 in Fig. 5. Figure 6 is the screen-shots of continuous jumping consisting of multiply jumping cycles. 1–6 in Fig. 6 show the robot can jump forward with controlled jumping speed and jumping height. And 7–9 in Fig. 6 show the robot jumps backward. The trajectories of the foot and the trunk are depicted in Fig. 7. The trunk trajectory is like a sinuous line, and the foot trajectory is also like a cyclic curve, which means that the jumping cycles are stable. Figure 8 shows the output torque from the ankle joint, the knee joint, and the hip joint, respectively. All of the three joints output torque is within the maximum torque the joints can exert. Summarizing, the vertical jumping and continuous jumping simulations verify the proposed controller as suitable for controlled jumping dynamics. 300 ankle knee hip

Torque(Nm)

200 100 0 -100 -200

0

1

2

3

4

5

Time(Seconds) Fig. 8. The torque of ankle joint, knee joint, and hip joint during robot jumping

5

Conclusion

In this paper, we propose a continuous jumping controller that is based on VMC for a one-leg robot platform. The virtual force and torque acting on the trunk are used to control the three states of a jumping cycle. An event based FSM is adopted to set the triggers among different jumping to realize continuous jumping. The simulations showed the effectiveness of the proposed controller. The results of this paper may provide a basis of fast running for bipedal robots.

Continuous Jumping Control

33

Acknowledgment. This work was supported in part by the National Natural Science Foundation of China under Grant 61903038 , 91748202, and 61533004, and in part by the 111 Project under Grant B08043.

References 1. Kaneko, K., Kaminaga, H., Sakaguchi, T., Kajita, S., Morisawa, M., Kumagai, I., Kanehiro, F.: Humanoid robot HRP-5p: an electrically actuated humanoid robot with high-power and wide-range joints. IEEE Robot. Autom. Lett. 4(2), 1431–1438 (2019) 2. Shigemi, S.: ASIMO and humanoid robot research at Honda. In: Humanoid Robotics: A Reference, pp. 1–36 (2018) 3. Hashimoto, K., Kang, H.-J., Nakamura, M., Falotico, E., Lim, H.-O., Takanishi, A., Laschi, C., Dario, P., Berthoz, A.: Realization of biped walking on soft ground with stabilization control based on gait analysis. In: 2012 IEEE/RSJ International Conference on Intelligent Robots and System, pp. 2064–2069. IEEE (2012) 4. Huang, Y., Vanderborght, B., Van Ham, R., Wang, Q., Van Damme, M., Xie, G., Lefeber, D.: Step length and velocity control of a dynamic bipedal walking robot with adaptable compliant joints. IEEE/ASME Trans. Mechatron. 18(2), 598–611 (2012) 5. Wainer, J., Robins, B., Amirabdollahian, F., Dautenhahn, K.: Using the humanoid robot KASPAR to autonomously play triadic games and facilitate collaborative play among children with autism. IEEE Trans. Auton. Ment. Dev. 6(3), 183–199 (2014) 6. Li, Q., Yu, Z., Chen, X., Zhou, Q., Zhang, W., Meng, L., Huang, Q.: Contact force/torque control based on viscoelastic model for stable bipedal walking on indefinite uneven terrain. IEEE Trans. Autom. Sci. Eng. 16, 1627–1639 (2019) 7. McMahon, J.J., Suchomel, T.J., Lake, J.P., Comfort, P.: Understanding the key phases of the countermovement jump force-time curve. Strength Conditioning J. 40(4), 96–106 (2018) 8. Raibert, M.H.: Legged Robots That Balance. MIT Press, Cambridge (1986) 9. LaViers, A.: Make robot motions natural. Nature 565(7740), 422–424 (2019) 10. Urata, J., Nakanishi, Y., Okada, K., Inaba, M.: Design of high torque and high speed leg module for high power humanoid. In: 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4497–4502. IEEE (2010) 11. Jiang, X., Chen, X., Yu, Z., Zhang, W., Meng, L., Huang, Q.: Motion planning for bipedal robot to perform jump maneuver. Appl. Sci. 8(1), 139 (2018) 12. Niiyama, R., Nagakubo, A., Kuniyoshi, Y.: Mowgli: a bipedal jumping and landing robot with an artificial musculoskeletal system. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pp. 2546–2551. IEEE (2007) 13. Hondo, T., Kinase, Y., Mizuuchi, I.: Jumping motion experiments on a NAO robot with elastic devices. In: 2012 12th IEEE-RAS International Conference on Humanoid Robots (Humanoids 2012), pp. 823–828. IEEE (2012) 14. Haldane, D.W., Plecnik, M.M., Yim, J.K., Fearing, R.S.: Robotic vertical jumping agility via series-elastic power modulation. Sci. Robot. 1(1) (2016) 15. Park, H.-W., Ramezani, A., Grizzle, J.W.: A finite-state machine for accommodating unexpected large ground-height variations in bipedal robot walking. IEEE Trans. Rob. 29(2), 331–345 (2012)

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System for Biped Humanoid Robot Focusing on Human Running Motion Hideki Mizukami1 , Takuya Otani2,3 , Juri Shimizu1,4 , Kenji Hashimoto3,5 , Masanori Sakaguchi6 , Yasuo Kawakami7 , Hun-ok Lim3,8 , and Atsuo Takanishi2,3(B) 1 Graduate School of Creative Science and Engineering, Waseda University, #41-304, 17

Kikui-Cho, Shinjuku-Ku, Tokyo 162-0044, Japan 2 Research Institute for Science and Engineering, Waseda University, Tokyo, Japan

[email protected] 3 Humanoid Robotics Institute, Waseda University, Tokyo, Japan 4 Hitachi, Ltd., Ibaraki, Japan 5 School of Science and Technology, Meiji University, Kanagawa, Japan 6 Institute of Sport Science, ASICS Corporation, Hyogo, Japan 7 Faculty of Sport Science, Waseda University, Saitama, Japan 8 Faculty of Engineering, Kanagawa University, Kanagawa, Japan

Abstract. We are developing the biped humanoid robot WATHLETE-1 (Waseda ATHLETE humanoid No.1) that has the same mass arrangement, link ratio, and output characteristics as humans. It is not possible to mount high-power electromagnetic motors and mechanical transmissions that satisfy the hip joint output at running because its weight makes it impossible to realize human body characteristics. To realize the characteristics of the human body, we adopted a hydraulic drive system that has a more advanced design than a mechanical transmission mechanism, that concentrates on the joints, and that can distribute output by splitting and merging oil. We propose a hydraulic circuit that improves the output of the actuator by connecting two hydraulic direct drive systems (HDDs) in parallel, independently mounted on the ankle and the hip joints using proportional valves. As a result, it has been confirmed that hip joint speed was improved by 200% compared to a single HDDs and with the potential of simulating the hip joint output required for a human running at 2.0 m/s. Keywords: Hydraulics · Parallel connection · Biped robot · Running

1 Introduction We have been developing a bipedal running humanoid WATHLETE-1 (Waseda ATHLETE humanoid No.1) (Fig. 1), which has the same mass arrangement and link ratio as humans, in order to explain human running motion from an engineering perspective. As a characteristic of the human running movement, it is known that running © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 34–42, 2021. https://doi.org/10.1007/978-3-030-58380-4_5

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System

35

legs function the same as a compression spring when standing [1]. The spring behavior of the knee and ankle joints is the same as a rotary spring [2], while the hip joints do not show much springiness at all. There is a moment when the ankle joint does not actively move during the stance phase, but it stores elastic energy in the elastic element of the joint and performs a kicking motion. In our previous research, we proposed a spring loaded inverted pendulum using pelvis (SLIP2 ) model [3], and we performed one-leg jumping using the elastic element and resonance mounted on the knee and ankle joints in addition to an electromagnetic motor. However, to satisfy the same mass arrangement and link ratio as humans, a highoutput electromagnetic motor and harmonic drive system that realize the output of the hip joint (Fig. 2) at a moving speed of 2.0 m/s, which the boundary at which humans transition from a walking motion to a running motion, cannot be mounted [4]. To solve this problem, we focused on a hydraulic drive to improve the output of the hip joint. The hydraulic drive can be expected to exhibit a high output similar to that used in Boston Dynamics’ ATLAS [5], which realizes running motion and parkour capabilities. Unlike the transmission mechanism using a reduction gear, the layout is good in that the hydraulic equipment can be placed at a location away from the joint axis itself.

Fig. 1. Whole body running robot, WATHLETE-1. The robot has 22 DOFs. A blue joint denotes the roll axis, a red joint denotes the pitch axis, and a yellow joint denotes the yaw axis in (b).

Hydraulic systems can share the output demonstrated by TaeMu, due to the diversion and merging of oil [6]. We have proposed an interlocking circuit that connects hydraulic direct drive system (HDDs) [7], which is mounted independently on the left and right, with solenoid valves. We selected equipment to achieve joint torque and angular velocity during a “human running at 2.0 m/s” test running and tested the proposed interlocking circuit. It was confirmed that the maximum possible thrust of the actuator had been sufficiently improved. Although the maximum sliding speed of the actuator had improved by 1.5 times, it was necessary to increase the speed by 1.8 times to achieve the required angular velocity of 4.5 rad/s. This is because the load on the interlocking cylinder and the solenoid valve had increased, and with the load becoming larger than anticipated. Therefore, to satisfy

36

H. Mizukami et al.

Fig. 2. Pelvis and hip joint mechanism equipped with electromagnetic motor and harmonic drive satisfying human mass arrangement and link ratio.

the required output, it is important that we would have to increase the size of the motor and hydraulic equipment, making it difficult to realize the mass characteristics and link ratio of humans. In this paper, we focus on the motion of the human ankle joint. Among the HDD systems that independently control the actuator speed with the pump discharge flow rate mounted on the lower leg of a biped humanoid robot, it is the ankle joint that is not driven during the stance phase. We propose a parallel connection circuit that can use the pump output to drive the hip joint. In addition, we constructed a device to evaluate the proposed system and assessed the effectiveness of the proposed method.

2 Consideration of the Proposed Hydraulic Circuit Figure 3 shows the transition of the joint angular velocities of the ankle and hip joints during one cycle of one leg that occurs during running motion. Table 1 shows the maximum joint torque and maximum joint angular velocity at the ankle and hip joints, respectively. In this anthropometric measurement, seven subjects ran at 2.0 m/s, over five trials. The movement of the human hip joint is periodic, and the phase is shifted by half a cycle on the left and right. For that reason, when the hip joint angular velocity is shifted by half a cycle (Fig. 3), the maximum exertion time of the hip joint angular velocity of the flexion motion in the swing phase and the extension motion in the stance phase of the right and left thighs roughly match. Therefore, it is necessary to slide the actuator at the maximum speed at both hip joints simultaneously. However, even if the hydraulic circuits of the left and right hip joints were connected, the size of the equipment would increase. Hence, we focused on the movement of the human ankle joint. As shown in the stance phase (Fig. 3), the ankle joint exhibits the maximum angular velocity from the time of landing (0.11 s) to the time of kicking out (0.38 s) during running. At that time, in addition to the active motion, it is considered that the passive motion where the impact of the landing is stored as elastic energy in the muscle of the calf and released at once is performed continuously. Focusing on the passive operation period of the ankle joint where it is not necessary to apply this drive torque, we examined improving the sliding speed of the actuator by connecting the ankle drive HDDs to the hip joint HDDs. We are studying the realization

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System

37

Fig. 3. Transition of joint angular velocity of hip joint pitch axis and ankle joint pitch axis when the human run at 2.0 m/s. The running leg has a stance phase and a swing phase. In the stance phase, there is a passive period in which part of the landing energy is stored in the calf as elastic energy.

of elastic energy accumulation by the ankle joint through a variable stiffness mechanism using leaf springs. Table 1. Joints output for human and previous robot Human 2.0 m/s WATHLETE-1 Pitch Axis Joint Ankle

Hip

Ankle

Hip

Max. Torque Nm

136

100

65

45

Max. Angular Velocity rad/s

5.9

4.5

4.6

2.9

Ankle Joint’s Valve5 Cylinder Valve1 Valve2

Link 2 Valve6 Valve3 Valve4

Link 1 Link 3 Link 4

Hip Joint’s Cylinder

M

Pump 1

Pump 2

Pump Unit Motor 1

Pump Unit Motor 2

M

Hip Joint Mechanism

Fig. 4. The proposed parallel connection circuit connects the HDDs circuit of the ankle joint and the HDDs circuit of the hip joint with valves 5 and 6. The hydraulic cylinder of the hip joint HDDs circuit is connected to a four-bar linkage mechanism that mimics the hip joint.

38

H. Mizukami et al.

Figure 4 shows the circuit to connect the HDDs for the ankle joint and the hip joint. The proposed circuit uses an HDDs that controls the meter-in flow rate to the cylinder that is controlled by the pump, and the meter-out flow rate from the cylinder that is controlled by the proportional valve. This circuit has additional proportional valves so that the HDDs for the ankle joint can be connected in parallel to the cylinder for the hip joint. In respect of the pipe resistance, the two circuits are connected at the cylinder port. Table 2 shows the parameters of the hydraulic circuit used in the test. Figure 5 shows the evaluation device. Table 2. Hydraulic circuit parameters Parameter

Value

Pump Displacement (cc/rev)

1.6

Pump Relief Pressure (MPa)

21

Pump Weight (kg/piece)

2.5

Cylinder Stroke (mm)

150

Cylinder Piston Diameter (mm)

25

Cylinder Rod Diameter (mm)

14

Cylinder Weight (kg/piece)

3

Variable Relief Valve Set Pressure (MPa)

16

Relief Valve Weight (kg/piece)

0.1

Hose Inner Diameter (mm)

6.4

Hose Length from Pump to Valve 1, 2, 3, 4, 5, 6 (mm)

300

Hose Length from Valve 1, 2 to Cylinder (mm)

1000

Hose Length from Valve 3, 4, 5, 6 to Cylinder (mm)

1500

Proportional Valve 1, 2, 3, 4, 5, 6 Nominal Flow Rate (L/min/MPa)

5/1

Proportional Valve Weight (kg/piece)

0.15

Oil Kinematic Viscosity at 40 °C (mm2 /s)

15

Oil Kinematic Viscosity at 100 °C (mm2 /s)

3.7

Experimental Oil Temperature (°C)

25

Motor Output (W)

200

Motor Weight (kg/piece)

1.5

Motor Rated Speed (rpm)

3000

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System

39

Fig. 5. The hip pitch axis mechanism mimics the left and right hip joint. In order to simplify the joint control, a link was used that linearly approximated the relationship between the joint angle and the cylinder stroke.

3 Evaluation of the Proposed Hydraulic Circuit Using the evaluation device, the maximum joint angular velocity and joint output by joint angular velocity and joint torque of the proposed circuit were evaluated. The results of a single HDDs circuit that controls the HDDs independently, the proposed hydraulic circuit that controls the HDDs of the ankle joint and the HDDs of the hip joint, and hydraulic interlocking circuit were compared. For the evaluation of the maximum angular velocity, the angle command step input was given to each hydraulic circuit without load. The command joint angle was given at 90° when the leg was driven from the vertical state to the horizontal state and when the leg was driven from the horizontal state to the vertical state. The speed when the cylinder was driven in the extension direction and the contraction direction is compared. In addition, the pump discharge pressure during operation and the inflow pressure of the cylinder were measured, and the pressure loss was compared. Figure 6 shows the results of the joint angle transition, angular velocity transition, and pressure loss transition of the hydraulic circuit. Evidently, among the three hydraulic circuits, the proposed hydraulic circuit converges to the command value fastest when the cylinder is expanded and contracted (Figs. 6a, 6b). When the cylinder is extended, the maximum joint angular velocity was 5.0 rad/s in the parallel connection circuit, which was two times that of the single circuit’s 2.5 rad/s (Fig. 6c). The joint angular velocity of the interlocking circuit was 3.8 rad/s (Fig. 6c). The pressure loss increased by 0.1 MPa in the parallel connection circuit and 0.5 MPa in the interlocking circuit compared to the single circuit (Fig. 6e). When the cylinder contracted, the joint circuit exhibited a joint angular velocity of 8.8 rad/s (Fig. 6d), which was 1.8 times that of the single circuit at 5 rad/s. In the case of the interlocking circuit, it was 6.9 rad/s (Fig. 6d). The pressure loss increased 0.7 MPa in the parallel connection circuit and 0.2 MPa in the interlocking circuit compared to the single circuit (Fig. 6f).

40

H. Mizukami et al.

Fig. 6. Each graph compares the individual HDDs circuit, hydraulic interlocking circuit, and proposed hydraulic circuit. The transition of the joint angle when a step input is given to change the hip joint mechanism from vertical to horizontal and from horizontal to vertical in (a) and, (b). The transition of the joint angular velocity when a step input is given to change the hip joint mechanism from the vertical state to the horizontal state and from the horizontal state to the vertical state in (c) and, (d). Pressure loss when the cylinder is extended and contracted in (e) and, (f).

For the evaluation of the joint output, weight was attached to the tip of the hip joint mechanism to observe the relationship between the available joint torque and the available joint angular velocity, and a step input was given in the same way as the measurement of the maximum joint angular velocity. The command angle was given at 90° from the vertical state to the horizontal state. The cylinder thrust force was measured with a load cell, and the joint torque was calculated from the relationship of the moment arm from the cylinder end to the joint axis. The weight was mounted in 10 kg steps, from 0 to 80 kg, and the output of each hydraulic circuit was compared (Fig. 7). It can

Investigation of Parallel Connection Circuit by Hydraulic Direct-Drive System

41

be seen that the proposed hydraulic circuit satisfies the joint output of a human running at 2.0 m/s. From the above results, we achieved the possibility that the proposed circuit can realize the hip joint angular velocity of a human running at 2.0 m/s. The factors that improved the speed were that there was no drive load on the idle leg side cylinder as in the interlocking circuit, and oil was merged before the inlet and outlet of the cylinder. With the pump drive torque rising slowly, it is believed that the discharge flow could be increased regardless of the motor torque upper limit.

Fig. 7. The relationship between the joint torque and the joint angular velocity of the three hydraulic circuits and the raw data of the hip joint output during human running at 2.0 m/s.

4 Conclusion In this paper, we proposed a hydraulic circuit that improves the sliding speed of the actuator using a parallel connection of a hydraulic direct drive system that controls the ankle and hip joints, referring to the transition characteristics of the legs during human running motion. In addition, an evaluation device that simulates the lower limb joint was constructed, and it was confirmed that the proposed hydraulic circuit increased the maximum angular velocity that could be demonstrated by two times compared to a single hydraulic direct drive system. By applying the proposed hydraulic circuit to a bipedal humanoid robot, we achieved the possibility of simulating the joint movement required for a human running at 2.0 m/s. In the future, we will develop a lower limb mechanism that satisfies the human link ratio and mass arrangement and evaluate its effect. The hip joint mechanism weighs 12 kg, but we plan to aim for a weight equivalent to that of a human by using a 3D printer and shape optimization. We evaluated using a general-purpose product, but we plan to reduce the weight by integrating the valve and actuator with the frame when mounting. Acknowledgements. Research was conducted with the support of Research Institute for Science and Engineering, Waseda University; Humanoid Robotics Institute, Waseda University; Human

42

H. Mizukami et al.

Performance Laboratory, Waseda University; Future Robotics Organization, Waseda University. It was also financially supported in part by the JSPS KAKENHI Grant No. 17H00767. Further, 3DCAD software SolidWorks was provided by SolidWorks Japan K.K.; cables and connectors were provided by DYDEN CORPORATION. We would also like to thank Editage (www.editag e.jp) for English language editing.

References 1. McMahon, T., et al.: The mechanics of running: how does stiffness couple with speed? J. Biomech. 23, 65–78 (1990) 2. Gunther, M., et al.: Joint stiffness of the ankle and the knee in running. J. Biomechan. 35, 1459–1474 (2002) 3. Otani, T., et al.: Utilization of human-like pelvic rotation for running robot. Front. Robot. AI, vol. 2, Article 17 (2015) 4. Thorstensson, A., Reberthson, H.: Adaptations to changing speed in human locomotion: speed of transition between walking and running. Acta Physiol. Scand. 131, 211–214 (1987) 5. Boston Dynamics. https://www.bostondynamics.com/atlas. Accessed 06 Feb 2020 6. Hyon, S., et al.: Design and experimental evaluation of a fast torque-controlled hydraulic humanoid robot. Trans. Mechatron. 22, 623–634 (2017) 7. Shimizu, J., et al.: Downsizing the Motors of a Biped Robot Using a Hydraulic Direct Drive System, Humanoids, pp. 580–586 (2018)

Maximal Output Admissible Set of Foot Position Control in Humanoid Walking Ko Yamamoto(B) , Ryo Yanase, and Yoshihiko Nakamura University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan {yamamoto.ko,yanase,nakamura}@ynl.t.u-tokyo.ac.jp

Abstract. This study describes the maximal output admissible (MOA) set simultaneously considering the COG stabilization based on the linear inverted pendulum model and the foot position control. The MOA set computes a feasible region for stabilizing controller and allows us to predict a falling risk against unknown external disturbances. However, the previous methods have a limitation that they cannot deal with the foot position control. The linear approximation of the support polygon constraint and the coordinate transformation in the control allow us to compute the MOA set of the foot position control. Keywords: Humanoid robot

1

· Foot position control · MOA set

Introduction

Push recovery is one of challenging tasks in the control of a biped humanoid robot. Because of limitations in an actual robot system, a robot needs to predict the falling risk beforehand. There are a lot of works on the falling risk prediction and the generation of an avoidance behavior. Based on a simplified model such as linear inverted pendulum (LIP) model [3], the capture point (CP) [6] was proposed. The feedback control of the CP maximizes the stabilizable region of the center of gravity (COG) balancing [7]. The model predictive control (MPC) is another effective method to generate a feasible biped walking motion, predicting the COG stability in future time-step [2]. Those approaches assume that the whole-body motion can precisely regenerate dynamics of the LIP model, including the property of a feedback gain in the LIP model. A feedback gain in the LIP model specifies the response of the zero moment point (ZMP) [8], which is an important indicator when predicting the falling risk. However, this is not the case in an actual robot because of the joint friction and compliance. Those factors change the response of the ZMP. As a result, the precision of the prediction would degenerate. One of the authors [10] utilized the maximal output admissible (MOA) set [1] for the falling risk prediction. In general, the MOA set is used for proof of stability of the MPC. It provides a set of states that guarantee the stability in This work was partially supported by JSPS KAKENHI Grant Number 18K19802. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 43–51, 2021. https://doi.org/10.1007/978-3-030-58380-4_6

44

K. Yamamoto et al.

pG -mg r

rs

c

fGRF pZ Fig. 1. LIP model of a biped humanoid robot

c r

x y

(a)

(b)

(c)

Fig. 2. Support polygon expression

the future time steps. Also, it has a good compatibility with the whole-body control [9]. However, the previous methods have a limitation that they cannot deal with the foot position control. In this study, we propose the MOA set simultaneously considering the COG stabilization and the foot position control. We show some simulation results based on the LIP model and verify the validity of the proposed method.

2 2.1

Control of COG Based on LIP Model and Its MOA Set LIP Model and State Feedback of COG

We consider a motion of a biped humanoid robot as shown in Fig. 1. Under the assumption that the total mass is concentrated on the COG, the motion equation of the COG is represented as follows:  fz ¨ G = ω 2 (pG − pZ ) − g, ω := (1) p m(zG − zZ ) where m is the total mass, pG = [xG yG zG ]T is the COG, pZ = [xZ yZ zZ ]T is the ZMP, f GRF = [fx fy fz ]T is the ground reaction force, and g = [0 0 g]T is the gravity acceleration vector. The LIP model assumes z¨G  0 and ω  const. The xy elements of (1) are extracted as a state equation form: x˙ = Ac x + B c u. The state x, control input u and state transition matrices Ac and B c are defined as follows:     T T   O E O x := xG yG x˙ G y˙ G , u := xZ yZ , Ac := 2 , B c := ω EO −ω 2 E

Maximal Output Admissible Set of Foot Position Control

45

where O and E are the zero and identity matrices with an appropriate size, respectively. Discretizing the state equation, we obtain the discrete-time system as follows: x[k + 1] = vAx[k] + Bu[k].

(2)

In this study, we focus on the tracking control of the COG to its reference. Let xref [k] and uref [k] denote references of x[k] and u[k], respectively. We suppose that xref [k] and uref [k] satisfy the state equation: xref [k + 1] = Axref [k] + Buref [k]. Then, we define the error vectors Δx[k] := xref [k]−x[k] and Δu[k] := uref [k]−u[k]. Hereafter, let Δ∗ denote the displacement of a physical quantity ∗ from its reference ∗ref : Δ∗ = ∗ref − ∗. By using those errors, the error dynamics is obtained as follows: Δx[k + 1] = AΔx[k] + BΔu[k]

(3)

This error dynamics is stabilized by the state feedback Δu[k] = −F Δx[k] where F is a state feedback gain. Substituting this into (3) yields the closed loop system  Δx[k + 1] = AΔx[k],

 := A − BF . A

(4)

From the recursive formula, x[k] is represented by using x[0] as follows:  k Δx[0] Δx[k] = A 2.2

(5)

System Constraint and Support Polygon Approximation

The constraint on the ZMP is that the ZMP exists inside the support polygon. In order to predict a falling, it is appropriate to consider a constraint that the CP exists inside the support polygon. Let z[k] denote a constrained variable. We suppose that z[k] is represented by a linear combination of the state and input vectors as follows: z[k] = Cx[k] + Du[k].

(6)

This equation can represent either the ZMP or CP by setting C = O, D = E or C = [E ω1 E], D = O, respectively. In a manner similar to (5), we consider the error of z as follows: Δz[k] = CΔx[k] + DΔu[k].

(7)

Substituting the state feedback Δu = −F Δx into (7) yields  Δz[k] = CΔx[k],

 := C − F D. C

(8)

The support polygon is defined as a convex hull of all contact points. Figure 2 shows the support polygon in the cases of (a) the single support (SS) and (b)

46

K. Yamamoto et al.

the double support (DS). A constraint that z[k] is inside the support polygon is represented as follows: M [k](z[k] − c) ≤ v[k]

(9)

where c is an inner point of the support polygon. The matrix M [k] and vector v[k] are obtained from the shape of the support polygon by computing a convex hull. This computation is nonlinear with respect to the change of the foot position, which makes it difficult to compute the MOA set of the foot position control. In this study, we approximate the expression of the support polygon. As shown in Fig. 2 (c), we linearly interpolate the left and right foot regions. First, we linearly interpolate between two foot positions r and r as follows:  s s r + r (s = 0, · · · , n) (10) rs = 1 − n n In the case of the DS, the positions of the left and right feet are selected as r and r. Then, we copy the foot sole region to each point r s as follows:

 Zs (r, r) = z ∈ R2 |M (z − r s ) ≤ v , (11) Finally, we approximate the support region by a combination of Zs as follows: Z(r, r) =

n

Zs = Z0 ∪ · · · ∪ Zn

(12)

s=0

The expression of (12) represents left foot support (LS) by Z(r L , r L ), DS by Z(r L , r R ) or Z(r R , r L ), and right foot support (RS) by Z(r R , r R ). Note that the above constraint is linear to both of z[k] and r s in any cases of the support state. This linearity of the constraint allows us to calculate the MOA set even when the foot position is changing. 2.3

MOA Set

The MOA set is defined as follows [1]: a set of initial states in which the constraint is satisfied with respect to a sequence of z[k] (k = 0, · · · , ∞). From (5) and (8), we can calculate z[k] by using the initial state Δx[0] as follows: k

A  Δx[0] z[k] = z ref [k] − C

(13)

From (11), (12) and (13), the set Xk of initial state error Δx that guarantees the constraint of z ∈ Z(r, r) at time k is obtained as follows: Xk (r, r) =

n s=0



  A  k Δx + r s ) ≤ v . Xk,s , Xk,s (r, r) = Δx ∈ R4 M (z ref − C

Maximal Output Admissible Set of Foot Position Control

47

Here, we assume that the foot positions do not change from their references, namely, r and r are constant. If Δx[k] ∈ Xk (r, r) is satisfied, it means that the constraint is satisfied at the current time step k. TheMOA set O∞ is obtained as a combination of Xk (k = 0, · · · , ∞), namely, ∞ O∞ = k=0 Xk . In the case of a regulator, O∞ is obtained with a finite number  as follows. O∞ =

 

Xk .

(14)

k=0

It is possible to analytically calculate the value of  [5]. x xref, i+1 i-th step

xref, i Support foot

x y

xref, i-1

k

Swing foot

Ti-1

N0 N1

N2

Ti

Fig. 3. Outline of planning a reference of biped gait

3 3.1

Integration of COG Control and Foot Landing Position Foot Position Control

Let Ti and r i+1 = [xi+1 yi+1 ]T denote time-step and the foot landing position at the i-th step in a walking motion. Figure 3 shows an outline of the i-th step motion, where the blue and green lines indicate trajectories of the ZMP and swing foot positions. The i-th step is defined between k = Ti−1 , · · · , Ti , where T0 = 0 is the initial time step of the total motion. Between the time steps, r i denotes the support foot position whereas the swing foot moves from r i−1 to r i+1 . In the next i + 1 step, the swing and support feet are switched. Let r[k] denote the swing foot position in the i-th step. As shown in Fig. 3, the boundary condition of r[k] is given as r[Ti−1 ] = r i−1 and r[Ti ] = r i+1 . To change the foot landing position r i+1 , we apply a simple linear feedback during swing motion as follows: r[k] = r ref [k] − G(xref [k] − x[k])

(15)

where G is a feedback gain matrix. In a manner similar to (3), this equation is rewritten by using the error of the foot position, Δr[k], as follows: Δr[k] = GΔx[k]

(16)

48

K. Yamamoto et al.

As a result of this control, the foot landing position is updated as follows: Δr i+1 = GΔx[Ti ]

(17)

In practical, (15) is applied during the swing motion, and we obtain (17) at the moment of the foot landing. 3.2

Control in Support Coordinate

The foot position control generates the error of the foot landing position as (17). When the error is large, this control is inconsistent with the COG control given by (4). To avoid this inconsistency, we reconsider the control of the COG and foot position in a coordinate whose origin is 1) the left foot position in LS, 2) the center of the feet positions in DS, and 3) the reft foot position in RS. For example, there are three phases in the i-th step in Fig. 3. In each phase, we consider a coordinate with the following origin point cij ∈ R2 (j = 0, 1, 2). ci0 = r, ci1 =

1 (r + r), ci2 = r 2

(a) without foot position control

(18)

(b) with foot position control

Fig. 4. Simulation result of the COG stabilization in the y-axis without/with foot position control

We call those coordinate as support coordinates. During Phase j in the i-th step (k = Ti−1 , · · · , Ti ), the coordinate transformations are summarized as follows: j

x[k] = x[k] − Scij , j u[k] = u[k] − cij , j z[k] = z[k] − cij , j r i+1 = r i+1 − cij

where the matrix S = [E O]T is an element-selection matrix from cij to x[k]. The system equations are rewritten in the j-th support coordinate as follows:  j x[k], Δj z[k] = CΔ  j x[k], Δj x[k + 1] = AΔ  j

j

j

Δ r i+1 = GΔ x[Ti ], Z(r, r) =

j

  n   j j z∈R  M ( z − rs ) ≤ v  2

s=0

Note that the above equations have the same expression with those in the original coordinate.

Maximal Output Admissible Set of Foot Position Control

3.3

49

Simulation of Foot Position Control

We validated the COG stabilization combined with the foot position control. In the following simulations, we generated a reference trajectory for a walking motion with total two steps, assuming a humanoid robot Hydra [4]. The reference trajectory was generated by using a method similar to [7]. For each step, the duration was 1 s, and the step length was 14 cm. The feedback gain F for the COG was given as the CP feedback [7]. Moreover, the feedback gain G for the foot position control is given as G = [E ω1 E]. This is an intuitive setting that the error of the foot position is determined by that of the COG. We compared the COG control with and without the foot position control. In both cases, the initial COG velocity in the y-axis was set to y˙ G = −0.3 m/s, assuming an impulsive disturbance. Figures 4 (a) and (b) show the results without and with the foot position control, respectively. In the case of Fig. 4 (a), the ZMP was not included in the support polygon when the second step motion started around 1.3 s. In the case of Fig. 4 (b), on the other hand, the landing position at the first step was moved in the −y direction by the foot position control, and the ZMP was included in the support polygon when the second step motion started.

4

MOA Set Considering Foot Position Control

As explained in Sect. 2, the computation procedure of the MOA consists of 1) n computing Xk,s , 2) Xk is obtained as Xk = s=0 Xk,s , and 3) MOA set O∞ is  obtained as O∞ = k=0 Xk . In 1), the following computations are required. a) Represent the state transition from Δx[0] to Δx[k] by Δx[k] = Ak Δx[0], where Ak is a state transition matrix.  k Δx[0] b) Represent z[k] by using Δx[0]: z[k] = z ref [k] − CA c) Represent r s by using Δx[0]: r s = r ref,s − Bk Δx[0] where Bk is another state transition matrix from x[0] to r s . Note that we can obtain the linear relationship between r s and Δx[0] because we assume the linear feedback in the foot position control (17).  k . When we consider In Sect. 2, we calculate the state transition matrix as Ak = A the foot position control with the coordinate transformation, the explanation of the computation of Ak and Bk requires more space of the paper. Therefore, we leave the computation of Ak and Bk for another paper. For a two-step walking motion with the COG stabilization and foot position control, we computed the MOA sets considering the ZMP constraint. Figures 5 (a) and (b) show the MOA sets without and with the foot position control. As shown in Fig. 5 (a), this reference walking motion has a very thin MOA set in the yG − y˙ G . As shown in Fig. 5 (b), on the other hand, much wider MOA set is obtained because of the foot position control.

50

K. Yamamoto et al.

(a) without foot position control

(b) without foot position control

Fig. 5. MOA set projected on the yG − y˙ G plane

5

Conclusion

We proposed the MOA set simultaneously considering the COG stabilization and the foot position control. A coordinate transformation in the control allowed a compatibility between the COG stabilization and the foot position control. Linearly approximating the support polygon constraint, we derived the MOA set as a linear inequality of the foot position. Simulation results based on the LIP model verified the validity of the proposed method. In future works, we validate that the MOA set can predict a falling risk, by whole-body dynamics simulations and actual experiments.

References 1. Gilbert, E., Tan, K.: Linear systems with state and control constraints: the theory and application of maximal output admissible sets. IEEE Trans. Automatic Control 36(9), 1008–1020 (1991) 2. Herdt, A., et al.: Online walking motion generation with automatic foot step placement. Adv. Robot. 24, 719–737 (2010) 3. Kajita, S., et al.: The 3D linear inverted pendulum mode: a simple modeling for a biped walking pattern generation. In: Proceedings of IEEE/RSJ IROS, pp. 239–246 (2001) 4. Kaminaga, H., et al.: Mechanism and control of whole-body electro-hydrostatic actuator driven humanoid robot hydra. In: Proceedings of ISER, pp. 656–665 (2016) 5. Kim, J., Lee, J., Oh, Y.: A theoretical framework for stability regions for standing balance of humanoids based on their LIPM treatment. IEEE Trans. Syst. Man Cybern.: Syst. 1–18 (2018) 6. Pratt, J., et al.: Capture point: a step toward humanoid push recovery. In: Proceedings of IEEE-RAS HUMANOIDS, pp. 200–207 (2006) 7. Sugihara, T.: Standing stabilizability and stepping maneuver in planar bipedalism based on the best com-zmp regulator. In: Proceedings IEEE ICRA, pp. 1966–1971 (2009)

Maximal Output Admissible Set of Foot Position Control

51

8. Vukobratovic, M., Stepanenko, J.: On the stability of anthropomorphic systems. Math. Biosci. 15(1), 1–37 (1972) 9. Yamamoto, K.: Humanoid motion analysis and control based on cog viscoelasticity. Adv. Robot. 31(7), 341–354 (2017) 10. Yamamoto, K.: Control strategy switching for humanoid robots based on maximal output admissible set. Robot. Autonom. Syst. 81, 17–32 (2016)

Control System Design for Human Assisting Robot Teresa Zielinska1(B) 1

and Michele Tartari2

Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Warsaw, Poland [email protected] 2 Department of Advanced Robotics, Istituto Italiano di Tecnologia, Genoa, Italy [email protected] https://ztmir.meil.pw.edu.pl/web/eng/Pracownicy/prof.-Teresa-Zielinska, https://m-tartari.eu/

Abstract. The paper presents study on robotics control system functions distribution using the concept of an embodied agent. Architecture of control system is specified and three options of system tasks distribution are tested considering the working efficiency. As an example the small, energy-efficient robot interacting with a human is used. Control system actions are designed aiming on a reliable human-robot interface. System serves the verbal and visual commands. Presented methodology and implementation tools are suitable and beneficial for the development of human assisting robots.

Keywords: Assisting robots interface.

1

· Control system · Human-robot

Introduction

The crucial part of an any autonomous robot is the control system. Creation of systems must involve a specification phase that defines the architecture, and an implementation phase that discloses how it was done. In advanced embedded robotics systems and in low cost simple robots with limited resources too, the control tasks distribution and/or grouping is very relevant for working efficiency. Human assisting robots shall be equipped with adequate interfaces. The end-user must be able to interact and operate the assisting machine regardless of his/her age, background and capabilities. Therefore, an easy to use human-robot interfaces are the imperative for the progress in the human assisting robotics. For a human, the voice represents the most natural means for delivering the orders. User-friendly interfaces should use voice recognition to receive and execute verbal commands. Despite of its complexity, the gesture that accompanies speech Supported by Faculty of Power and Aeronautical Engineering, Warsaw University of Technology. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 52–59, 2021. https://doi.org/10.1007/978-3-030-58380-4_7

Control System Design for Human Assisting Robot

53

also passes the information. Trained specialists can get substantive information from gestures – information that is not always identical to that gleaned from the speech [4]. Consideration of gesture semantics would not only further simplify the commands perception with increasing the accuracy with which robots understand humans, but it would also limit errors by detecting passed information contradictions and will prevent of misunderstandings of the user. Properly designed system architecture enhances the interfacing efficiency. It concerns the time-responsiveness of the system and the reliability of communication. Even the most efficient human-robot interface can not perform sufficiently due to the weaknesses of control architecture. Such observation leads to the topic of this paper. The remaining part of this paper is organized as follows: state of the art, research objective and robotic system is described. The human-robot interface and functional architecture of the control system are presented next, three options of tasks distribution are proposed. The final part describes shortly the test results, and the best solution is indicated. The paper is ending with conclusions.

2

State of the Art

The term of robot control system architecture pertains to the division of the system into subsystems and their interconnections, whilst the activities are specified by a set of formal concepts as: Finite State Machines (FSMs), behaviours, transition functions etc. with inter subsystem communication means. Proper design of software architecture determines system reliability and flexibility for farther upgrades and modifications [6]. The decisions about the architecture must be taken in the early stage of the developmental work. It is a very important but difficult task considering that the decisions have the far-reaching effect. Besides of defining the architecture, the proper design of the human-robot interfacing, is especially relevant for human assisting robots. The type of human-robot interface (HRI) depends on expected proximity [3] which can be: – remote: the human and the robot are not co-located and are separated spatially or even temporally (e.g., the Mars Rovers), – close: the humans and the robots are sharing the environment. The HRI are using following communication media [8]: • physical: the human is physically coupled to, and cooperating with, the robotic device. We can have direct physical contact between human and robot, or their physical interaction can be mediated through a third object. The robot must be equipped with adequate force and touch sensors [5], • visual: the robot is capable of interacting with humans by analyzing input from cameras and/or by displaying content on one or more screens [1], • vocal: the robot is capable of analyzing and/or produce sounds in a way that allows it to interact and coordinate with humans, it often requires speech recognition and/or generation, • biological: the robot is properly equipped to receive, read, and understand biological signals such as electroencephalographic (EEG) or electromyography (EMG) signals [2], • mixed: the robot is capable of interacting with humans using two or more communication channels [7]. Traditional interaction utilities,

54

T. Zielinska and M. Tartari

such as screens, buttons, and touch screens unfortunately involve rather complex menu structure. Using those utilities while performing a complex spatial task decreases the overall performance as this type of interaction competes for limited resources. Thus, it has been favored a vocal interface for the assisting robots. This implies that the robot must have the capabilities to understand and interpret the human speech what was considered in presented research.

3

Objective

Basis on the state of the art, the aim of presented research is the design of an efficient tasks distribution within the control system considering the needs of established HRI which uses verbal and visual communication. The solution is dedicated for un-expensive mobile robots performing some simple tasks according to human commands. As one of such tasks, transportation of some goods along the path shown by a human in the form of visual of voice command can be considered. Taking into account the needs of human assisting robotics the structure of control system was elaborated with the stress laid on the reliable interfacing with the human being. The vocal interface has been then brought to a deeper implementation level by considering together the visual and vocal information. It was assumed that the system must be cheap, avoiding the high computation facilities. For this purpose the cheap testing robot was developed. The robot (Fig. 1A) with two actuated wheels is equipped with three IR sensors detecting obstacles in the range 3–40cm. The hardware components are: Raspberry Pi V2.1 camera, an Intel Realsense depth camera, a Raspberry Pi 3B on-board computer, and Arduino Micro microcontroller board. The Bluetooth headphone allows to give the commands to the robot. In the testing stage the results are displayed on the laptop screen. The joystick is used as a safety measure for gaining the direct robot control when needed. Despite of the low weight and small dimensions, the system retains enough capabilities for serving a human being. Arduino is responsible for the low-level control of the actuators and for interfacing with the sensors. The Raspberry Pi, running Ubuntu Mate 16.04 and ROS kinetic holds the main control system with the subsystems. The data from the sensors are there processed and the commands for the effectors are produced. Project was been realized using ROS with the assumption that the software must be as much as possible hardware independent allowing easy upgrades and replacements of hardware components.

4

Human-Robot Interface

In presented work the stress is laid to the control system design considering the needs of HRI. The verbal and visual interface was established in a simple but sufficiently useful way. It was assumed that the user wears the yellow jacked. Such assumption allows an easy recognition of the user in crowded environment. The information about the user pose is obtained from the processed image. The RGB image is first transformed into the high saturation space (HSV) format, then the

Control System Design for Human Assisting Robot

55

Fig. 1. Illustration of the robot and the control system design. The developed robot (A). Behaviors selection (B). Functional architecture of the system: first option of tasks distribution (C), second option of tasks distribution (D), third option of tasks distribution (E).

software checks whether pixels are in the appropriate color range. Accepted color area is adjusted within the rectangle making the assumption that it represents an user. Next the system follows the center of identified rectangle and through the mapping, transforms an user position into the depth image. The estimation of the user position with respect to the robot is sufficiently accurate. All elements of the image which are farther than the detected distance to the user are neglected. The image is processed delivering the user posture what constitutes the visual command. It is expected that the human expresses the command concurrently by the upper limbs position (visual command) and by the voice (Fig. 2).

56

T. Zielinska and M. Tartari

Fig. 2. Image processing for user detection.

5

General Architecture of Control System

The general scheme of functional architecture of the control system together with discussed later three options of the tasks distribution is shown in Fig. 1. The global navigation part is responsible for serving user requests passed verbally and visually, and defines the target position for the robot. It must be noted that the global navigation not appears as a single element but is split to a set of three subsystems, namely the voice interface, the user detector, and the executioner. After receiving the target position, the local navigation defines required robot’s path and sends the velocity commands to the actuation layer controlling the actuators. The safety module ensures a risk-free operation around humans, this subsystem inhibits lower subsystems and takes the direct communication with the actuation control any time when risks occur. The executioner as the main part of the system, is responsible for creating the robot behaviors. Its action is described by the states selector shown in Fig. 1B. The following states are defined: Bidle – the initial state, passive waiting. This state is entered anytime when the “stop” command is received, or inhibition is reported, this state is left only after the inhibition is ended, and a valid voice command is received, Bconf lict – this state is entered every time when the voice command is contradictory to the visual information (established by the hands position) produced by the user (e.g. “turn left” while showing the right direction). The execution process issues a voice reply to notify the user about confusing information and then the process transfers to Bidle , BdiscreteM otion – this state is entered anytime when the discrete motion is requested (e.g. “turn left”) with no contradiction with the user visual command, the robot keeps moving until the inhibition or until a new voice command is given, or until the time limit runs out, whatever occurs first, Bf ollowT arget – this state is activated by receiving the voice commands “rotate left” or “move back”, and its functioning is very similar to the one of BdiscreteM otion with the main difference that no time limit is used. Bf ollowU ser – this is a kind of super-state (we address it as the behavior) is activated by the voice command “follow me”. The work starts with the S wait , where the robot stands still and waits until the user is detected in the cameras field of view. When it happens, it goes transition to S f ollow state, a motion is requested and the robot moves and place itself facing the operator in the fixed distance from him/her. If the user moves, the robot adjusts its pose accordingly, when the user stays still the subsystem is in the state S wait .

Control System Design for Human Assisting Robot

6 6.1

57

Detailed Architecture Division of System Tasks

The concept of embodied agent helps to divide the workload between the parts of the robotic control system. The embodied agents retrieve information from receptors, elaborate it to take the decisions, and finally modify the surrounding environment through the effectors [9]. Agent αi is described by the four-tuple: (C, E, R, T ), where C represents the control action, E represents the interaction with the ambient through sensors, R stands fort the capacity to perceive through receptors, and T represents inter-agent communication (transmission). Any agent needs the C component to realize its functions and can have some other components of the four-tuple [9]. The C, CE, or CR agents are of limited utility for robotics. In our work a three-agent system has been developed by gradually assigning the user interfaces to one or more agents. In general, we investigated the relation between distribution of the (C, E, R, T ) actions between the agents and the system performance. Three options were designed, implemented and tested. The first tested was the most centralized structure among three investigated. Here the display agent αd is of type CET , and can be considered as a mere virtual effector. This agent displays processed images and it receives information from the control agent. The control agent is of latter is of CERT type, and it is responsible for the voice interface, user detection, navigation, and

Fig. 3. Testing some of the actions defined for user following task. Conflicting commands – robot not moves (1). “Follow me” – the robot is ready to move (2). Robot is following the user motion (3). User stopped, the robot stopped facing the user (4).

58

T. Zielinska and M. Tartari

safety. Figure 1C shows the structure of these two agents in this option. The second option is more relaxed structure. However the control agent still retains most of the functions. It is composed of the executioner, the user detection, navigation, and the safety module. However, the voice interface is now managed by the display agent, which processes the audio stream and sends the voice commands in a string form. Using this structure, all agents are of CERT type (Fig. 1D). The third and final option has more equally distributed computational tasks among the system when comparing to the previous two. The display agent besides of showing processed images, governs also the voice recognition and safety module what makes the executioner, user detection, and navigation only the control agents. Figure 1E shows the display and the control agent. They are of type CERT and CRT respectively. Each option was implemented and evaluated. 6.2

Selecting the Most Efficient Solution

Several test cases and scenarios were executed, some examples are illustrated in comprehensive way by Fig. 3. The tests were performed not only by one, but by several persons to verify that it is not specific user dependent. Taking into account that the robot is intended for interaction with humans, the comparison of different tasks distributions (architectures) has taken place on a qualitative level rather than using pure performance benchmarks. The first option (Fig. 1C) gave good results considering the responsiveness to the commands, but occasionally some other system performance were dropping due to the high CPU and RAM usage (both over 90% of their capabilities). Such performance deterioration had a short duration and did not affected the overall functioning of the system, it was only perceivable by a lower refresh rate of the processed images and slightly higher time required to react to a given command. The second option (Fig. 1D) allowed a more complex model for speech recognition, which resulted in more precise commands detection while maintaining stable performances throughout the whole usage of the robot. This came to the cost of a bit larger delay between the moment when a command was given and the instant in which it was executed. The Raspberry Pi resource usage remained at acceptable levels (60–80% of CPU usage and 80% of RAM usage). For the current hardware setup, the best results were obtained for this implementation. In fact, further distribution of the computational load in the third option (Fig. 1E) didn’t provided any major improvement in the system performances. At the same time, an increased reaction delay due to the higher amount of data transferred between agents was observed. The command contradictions are detected by comparing the voice commands with the visual information extracted from the body posture.

7

Conclusions

The control system architecture was proposed. Three options of the system tasks clustering within the system agents were implemented. Basis on the tests performed on the robot equipped with carefully selected HRI, the most efficient tasks

Control System Design for Human Assisting Robot

59

distribution was indicated considering communication needs of human assisting robots. The final design of control system satisfies the requirements, both – in the terms of usability and performances. Identified limitations were only cased by applied hardware, namely: – the USB2.0 applied for communication with the camera limited data transmission and prevented using advanced vision capabilities, – the simple, low powered board (Raspberry Pi) was putting the limit to the amount of processed data, – small memory prevented of adding more sensors therefore enhancing the system capabilities (such as self-localization). The future work will focus on the optimization of the memory handling using the simplewhiteboard library, and on the addition of new actuators, sensors, and software components establishing the new functions of the robot such as self-localization and object identification. The design and implementation of the overall system was made by M. Tartari advised by T. Zielinska.

References 1. Carcagni, P., et al.: Visual interaction including biometrics information for a socially assistive robotic platform. In: European Conference on Computer Vision, pp. 391– 406 (2014) 2. Bu, N., et al.: A hybrid motion classification approach for EMG-based human–robot interfaces using bayesian and neural networks. IEEE Trans. Robot. 24, 502–511 (2009) 3. Goodrich, M.A., et al.: Foundations and trends. Hum. Comput. Interact. 1(3), 203– 275 (2008) 4. Goldin-Meadow, S.: The role of gesture in communication and thinking. Trends Cogn. Sci. 1(11), 419–429 (1999) 5. Heinzmann, J., Zelinsky, A.: Quantitative safety guarantees for physical humanrobot interaction. Int. J. Robot. Res. 22(7–8), 479–504 (2003) 6. Hussain, R., Zielinska, T., Hexel, R.: Finite state automaton based control system for walking machines. Int. J. Adv. Robot. Syst. (2019). https://doi.org/10.1177/ 1729881419853182 7. Hiroshi, O., et al.: Human-robot non-verbal interaction empowered by real-time auditory & visual multiple-talker tracking. Adv. Robot. 17(22), 115–130 (2003) 8. Yan, H., Ang, M., Poo, A.N.: A survey on perception methods for human-robot interaction in social robots. Int. J. Soc. Robot. 5(1), 85–119 (2014) 9. Zielinski, C., et al.: Agent-based structures of robot systems. In: Trends in Advanced Intelligent Control, Optimization & Automation, pp. 493–502. Springer, Cham (2017)

Development of a Switchable Wearable Robot for Rehabilitation After Surgery of Knee Koji Makino1(B) , Teppei Ogura1 , Masahiro Nakamura2 , and Hidetsugu Terada1 1 2

University of Yamanashi, Kofu Takeda 4-3-11, Yamanashi, Japan [email protected] Kofu Municipal Hospital, Kofu Masuhira 366, Yamanashi, Japan

Abstract. It is important to give the patient to the suitable rehabilitation after almost surgical operations, however it is hard work for the physical therapist and patient. Therefore, the robot for rehabilitation is desirable. We developed the wearable assist robot for rehabilitation after the Total Knee Arthroplasty (TKA) surgery. In our robot, two type robots for left leg and for right leg are necessary, and cost for the introduction in a hospital becomes high. The other similar robots for rehabilitation are not distinguished into each leg, therefore the weight is larger and it is difficult for patient to walk freely. To solve the problem, the new switchable robot employed the bilateral asymmetrical gear for the knee joint and the new nondirectional brace to fix the robot to each leg is developed. And the effectiveness of the new robot is confirmed by the sensory evaluation, myoelectricity and movement of the brace using the developed real switchable robot.

Keywords: Rehabilitation robot Nondirectional brace.

1

· Bilateral asymmetrical gear ·

Introduction

The disease or injury of the knee influences on maintenance of a QOL (Quality of Life), since the patient is difficult to walk. Therefore, the patient is given a surgery of knee, if disease or injury of the knee occurs. Especially, many elderly persons have osteoarthritis due to wear of cartilage not only injuries. The most effective surgery for knee osteoarthritis is Total Knee Arthroplasty (TKA), which has contributed to improved QOL. The number of the surgical operation of the TKA is 700,000 cases in the United States. After the surgery, the patient is necessary to perform a rehabilitation to modify the motion of walking. If the motion is not modified, the doctor and the physical therapist think that the patient stumbles on a small step and that the other knee becomes worse. Therefore, the patient is necessary to receive the rehabilitation. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 60–67, 2021. https://doi.org/10.1007/978-3-030-58380-4_8

Development of a Switchable Wearable Robot

61

Almost of the rehabilitation are performed by the physical therapists. There is two kind of the rehabilitation performed by the physical therapist; one is the ROM (Range of Motion) training, the other is the walking rehabilitation. The ROM training is the motion for improving the flexibility and stretching of the joint, and is performed by the physical therapist. A lot of the instruments of the training are studied [1,2] and developed [3,4] also known as CPM (Continuous Passive Motion)[5] to decrease the task of the physical therapist. On the other hand, the patient walks using a carry cart or a stick for the walking rehabilitation, and the physical therapist gives the advice to the patient to improve the motion of the walking. It is important to walk practically. The equipment type robots are studied [1–4], which move the knee joint of the patient automatically, therefore it is unnecessary for the patient to move the knee on own opinion. In the walking rehabilitation, it is important to move the knee on own opinion. Therefore, we have developed the wearable assist robot for walk rehabilitation after the TKA [6] that enables the patient to walk on own opinion and to improve the motion of the walk by the weak assist force. The robot is enough to support the assistive motion of the one knee joint, since only one leg is often given to the surgery. Therefore, two robots for left leg and right leg are necessary, and the hospital must buy two robots, if our developed robot is introduced. It is desirable that a robot can be attach to the each of legs. This paper describes the new robot that is able to be switched for each leg. The cost to introduce it in a hospital is half, if the robot is realized. The Sect. 2 describes the previous developed robot and difficulty to switch the leg. Next, the new switchable robot is shown in Sect. 3. And the result of experiments using the real new robot is shown in Sect. 4. Final, Sect. 5 is conclusion.

2 2.1

Previous Developed Robot Shape of the Previous Robot

The previous developed robot is illustrated in Fig. 1 In this robot, the same shape parts are equipped with the both legs. However, one leg part includes the actuator, and the other leg part does not has the actuator and has the battery. There is a problem in a cost, since both type for left leg and for right leg is necessary to introduce the robot in a hospital. To solve the problem, we think that both leg parts include the actuators. There are two problems in this case. One is that the muscle of the knee that is not given the surgery becomes weak by the assist for the knee, if the assist is added. The other is that the weight is larger than the robot, if one of two actuators is not operated. It is difficult to modify the previous robot to a switchable robot. 2.2

Roll-Back Motion

The roll-back motion of the knee is explained using Fig. 2(a). The motion of the knee joint is not simple rotation type, and it consists of rotation and slide motion.

62

K. Makino et al. Imaginary center Roll back

Femur

0 degrees

(2)

(3)

(1)

Roll back

Patella

Contact point 60 degrees

Tibia

Roll back

(3) (2)

(1)

120 degrees

(a) bone

Fig. 1. The previous robot.

(b)the prev. gear

Fig. 2. Roll-back motion

We developed the asymmetrical gear corresponding with roll-back motion as shown in Fig. 2(b). The previous developed robot adapted the asymmetrical gear. Therefore, the smooth motion of the knee is realized. However, it has a problem. The opposite side motion cannot be realized, since the tooth of the gear cannot be generated because of the pinion gear and the bar to be connected with the leg part. Therefore, the previous robot for left leg cannot be attached to opposite side leg, since the direction of motion is inverse in left and right side. It is difficult to employ the previous robot to each leg without simple modification.

3 3.1

New Switchable Robot Concept of the Robot

The robot is enough to support the assistive motion of the one knee joint, therefore one leg part is necessary as shown in Fig. 3. The patient “A” equips the robot attached to the left leg as shown in Fig. 3. Assume that it is released. The other patient “B” equips the same robot on right leg. In the previous robot, the switching action cannot be executed. 3.2

Bilateral Roll-Back Gear

We have proposed the mechanism for bilateral roll-back motion on knee joint [7]. The mechanism is explained briefly. This gear has non-circular gear and two grooved cams as shown in Fig. 4 as similar to Fig. 2(b). In this gear, the center locus of grooved cam is not coincident with the locus of the knee joint’s imaginary rotation center, and the driving gear meshed with the non-circular gear drives the lower leg side plate, and the plate guided by two grooved cams rotates with a roll-back motion. Figure 4 shows the bilateral motion of the gear.

Development of a Switchable Wearable Robot

3.3

63

Non Directional Brace

It is important to fix the robot to the patient rigidly. In the previous robot, the shape of the brace that means the device to fix the leg part to the thigh or calf is two kind as shown in Fig. 5(a); one is for left leg and the other is for right leg. They are asymmetrical shapes, since the hook-and-loop fastener is closed on front of leg. In the new robot, the shape is modified to half circle type, and the special hook and loop fastener with elastic property is employed as shown in Fig. 5(b). The attachment motion is the same, whether the leg part is fixed to the left or right leg (Fig. 4).

rem oval

inversio n

equipm ent

P atient A

P atient B

Fig. 3. The concept of the new switchable robot.

- 120 degrees

- 60 degrees

0 degrees

60 degrees

1 2 0 degrees

Fig. 4. The bilateral roll–back gear and the motion.

4

Evaluation of the Switchable Robot

The new switchable robot is developed as shown in Fig. 6. And the weight of the previous and new robot are 6.73 kg and 4.95 kg, respectively. In this section, the robot is evaluated by three methods.

64

4.1

K. Makino et al.

Sensory Evaluation

The sensory evaluation by questionnaire is investigated. The questionnaire items are following; motion(easy–difficult), balance(good–bad), and weight(heavy– light), and each score is five rank. The subjects are three men, 22–24 years old, and don’t have a disease or injury of the leg.

(a) Previous brace

(b) new brace Fig. 5. The shape of the previous brace is asymmetrical, and the new brace can fix the robot by the same action since the shape is symmetrical.

The average score of each factor of the previous robot and the new robot is following; motion scores are 2.6(prev.) and 4.2(new), balance scores are the same 3.4 and weight scores are 2.4(prev.) and 4.2(new). And sum of the scores are 8.4(prev.) and 11.8(new). And, the comment is as follows: – A subject feels that the weight of the new robot is lighter. – The motion of walking is more natural using the new robot. – It is easier to rise up the thigh using the new robot. As a result, the new robot is better in total feeling judgement than the previous robot.

Development of a Switchable Wearable Robot

(a) left

65

(b) right

Fig. 6. Shape of the new switchable robot.

4.2

Myoelectricity

The myoelectricity is measured to investigate whether the shape of the robot is influence on the motion of the leg [8]. The subject is one among the subjects who participated in the questionnaire. The measurement of muscles are vastus medialis muscle that is used to stretches the knee joint and gluteus maximus muscle that stretches the hip joint. The results are shown in Table 1 and 2. Table 1 shows the myoelectricity of the vastus medialis muscle (VMM) and gluteus maximus muscle (GMM) using the previous robot and the new robot in case that the left leg is assisted by the actuator. On the other hand, Table 2 shows the myoelectricity, when the right leg is assisted. The average value of the myoelectricity is shown. And the ratio calculated by the division of the value of the electricity of the leg that equips the robot included the actuator by the value of the other leg is shown. We focus on the ratio. The value of the ratio are almost one. However, the values of the ratio the GMM using the new robot are larger. The myoelectricity of GMM influences on the motion of the hip joint. Therefore, it is clear that the leg motion using the new robot is larger than one using the previous robot. As a result, the advantage of the new robot is confirmed. Table 1. It is equiped to left leg. VMM prev. right 35.6 left 33.7 ratio 1.06

4.3

(knee) new 35.4 33.6 1.05

GMM (hip) prev. new right 3.8 4.8 left 5.9 3.9 ratio 0.64 1.23

Table 2. It is equipped to right leg. VMM (knee) GMM (hip) prev. new prev. new right 35.4 35.6 right 5.3 2.9 left 33.5 33.8 left 5.1 5.5 ratio 0.94 0.95 ratio 0.95 1.90

Movement of the Brace

Finally, the movement of the brace of the previous and new robot is compared, when the subject walks. The amount of movement is measured by the movie

66

K. Makino et al.

4

4

3

3

2 1 0 -1 0

2

4

-2

6 Tim e [s]

(a) prev.

8

10

O ffse t [m m ]

O ffse t [m m ]

camera. The results are shown in Fig. 7. The maximum movement of the brace using the previous robot is 3 mm, and one using the new robot is 2 mm. As a result, we confirm that the equipment feature of the new robot is higher.

2 1 0 -1 0 -2

2

4

6

8

10

Tim e [s]

(b) new

Fig. 7. The amount of the offset of the brace.

5

Conclusion

This paper describes the new switchable robot without complex procedure to change the leg. The merits of the new robot enable the patient to reduce own load in rehabilitation, and enable the cost for introduction of the robot in the hospital to decrease. To realize the new robot, the previous robot is improved in two points. One is to adopt the bilateral directional gear, the other is to modify the brace. Furthermore, the new robot is developed in fact and is evaluated in the sensory evaluation, myoelectricity and movement of the brace. As a result, we confirm the effectiveness. Now, we conduct clinical tests using the previous robot. In a few month, the previous robot will be replaced to the new switchable robot. And the effectiveness for the patients will be investigated.

References 1. Ogata, T., et al.: Hybrid Assistive Limb (HAL) rehabilitation in patients with acute hemorrhagic stroke. Neurologia Medico-Chirurgica 55(12), 901–906 (2015). https:// doi.org/10.2176/nmc.oa.2015-0209 2. Colombo, G., Wirz, M., Dietz, V.: Driven gait orthosis for improvement of locomotor training in paraplegic patients. Spinal Cord 39, 252–255 (2001). https://doi.org/10. 1038/sj.sc.3101154 3. Vita, D.P., et al.: A functional knee brace alters joint torque and power patterns during walking and running. J. Biomech. 29(5), 583–588 (1996). https://doi.org/ 10.1016/0021-9290(95)00115-8 4. Antonio, J., et al.: Hybrid therapy of walking with Kinesis overground robot for persons with incomplete spinal cord injury: a feasibility study. Robot. Auton. Syst. 73, 44–58 (2015). https://doi.org/10.1016/j.robot.2014.10.014

Development of a Switchable Wearable Robot

67

5. Anton, F.L., et al.: Effectiveness of prolonged use of continuous passive motion (CPM) as an adjunct to physiotherapy following total knee arthroplasty: design of a randomised controlled trial. BMC Musculoskeletal Disorders 7(15), 1–5 (2006). https://doi.org/10.1186/1471-2474-7-15 6. Zhu, Y., et al.: Study of wearable knee assistive instruments for walk rehabilitation. J. Adv. Mech. Des. Syst. Manuf. 6(2), 260–273 (2012). https://doi.org/10.1299/ jamdsm.6.260 7. Terada, H., Makino, K., Ishida, K., Ogura, T.: Development of a knee joint assistivemechanism adapted for bilateral roll-back motion. In: Proceedings of the 7th European Conference on Mechanism Science, Mechanisms and Machine Science 59, 19–26 (2018). https://doi.org/10.1007/978-3-319-98020-1 3 8. Erni, T., Colombo, G. : Locomotor training in paraplegic patients: a new approach to assess changes in leg muscle EMG patterns. Electroencephalography Clin. Neurophysiology/Electromyography Motor Control, 109(2), 135–139 (1998). https://doi. org/10.1016/S0924-980X(98)00005-8

Design and Testing of BIT Flying Robot Yunqi Liu1(B) , Long Li1,2 , Marco Ceccarelli2,3 , Hui Li1,2 , Qiang Huang1,2 , and Xiang Wang2,4 1 Beijing Institute of Technology, Beijing 100081, China

[email protected] 2 Beijing Advanced Innovation Center for Intelligent Robots and Systems,

Beijing Institute of Technology, Beijing 100081, China 3 LARM2: Laboratory of Robot Mechatronics, Department of Industrial Engineering,

University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy 4 China Academy of Space Technology, Beijing 100081, China

Abstract. Flying robots can be used for monitoring and maintenance in space orbital stations to check continuously even places not easily reachable by astronauts. The BIT flying robot is presented with its features for such an operation with small and versatile spherical structure. Basic performance characteristics are validated in lab experimental tests simulating no gravity. The result shows feasibility and peculiarity of the BIT flying robot. Keywords: Flying robots · Design · Performance evaluation · Testing

1 Introduction As the deepening of human research on space, astronauts’ workload on the space station is gradually increasing. To ensure safety of astronauts and to reduce daily workload of astronauts, the United States, Japan, China and many other countries research flying robots to help astronauts to monitor the air quality in the space station, to detect the status of space station and to determine location of faults. NASA has developed 5 models of PSA (Personal Satellite Assistant) flying robots whose diameter is larger than 150 mm, [1]. These robots use LCD display, blowers and flywheels for motion control and attitude adjustment. In 2013, NASA designed a new Smart SPHERES that has been tested and experimented on the space station, [2]. The robot uses carbon dioxide for propulsion and communication is operated with a programmable Android phone. In this way, it enables low-cost expansion and rapid development of new features. Japan Aerospace Exploration Agency designed Int-Ball spherical robot of 150 mm in diameter with 12 micro fans in the propulsion unit, and a flywheel together serving as a attitude control unit, [3]. In China, Shenyang Institute of Automation designed a spherical robot [4] whose model with a Newton-Euler algorithm. The robot uses four fans for propulsion and attitude adjustment. The institute designed also a new-generation prototype AAR-2 with 12 fans as propeller, that is used also for attitude and motion adjustment. To ensure rigidity and to reduce mass, the robot is made of Aluminum alloy, [5]. The robot designed © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 68–75, 2021. https://doi.org/10.1007/978-3-030-58380-4_9

Design and Testing of BIT Flying Robot

69

by the National University of Defense Technology in China uses 12 miniature fans and flywheels for propulsion and attitude adjustment in a sphere of diameter 200 mm, [6]. In this paper, a novel spherical flying robot is presented as designed at BIT (Beijing Institute of Technology) with characteristics of motion decoupling for an easy control, the structure of small size and modular design. Operation performance are characterized by testing results to show the feasibility and peculiarity of the proposed design of BIT Flying Robot.

2 Design Issues and Requirements Due to the size limitation in orbital space stations, the application environment is quite limited as shown in Fig. 1 a). It is necessary to make a flying robot small and light with very variable operation modes. In Fig. 1 a), a flying robot scans the entire space station and monitors its state. That means the earlier any problems are found during the inspection, the earlier the astronauts can repair them.

a)

Real-time cabin image display

Video module

Real-time environment detection

Sensor module

Autonomous

Attitude module/ Thrust module

Motion driving

Control Unit

b)

Fig. 1. Design issues and requirements: a) a virtual representation of the working environment; b) a scheme.

Referring to Fig. 1 a), a flying robot is conceived with four basic modules, namely a video module for real-time image transmission, a sensor module for real-time environmental monitoring, a power module to ensure normal operation, and driving module as control unit. The status of a space station needs to be constantly monitored to detect any problem at any time, such as fire, damage, and other failures. In Fig. 1 b) a requirement for a visual monitoring of the environment status can be solved with a video system in a suitable small module on board. For detecting environment parameters, such as temperature, air quality, pressure and other status characteristics, a large number of sensors can be installed within a sensor module as distributed in the spherical volumes of the BIT Flying Robot. The autonomous operation with suitable planning can be achieved with adjustable motion that can be generated by attitude module and thrust module. However, astronauts can switch between automatic and manual driving modes for the robot operation through the control unit.

70

Y. Liu et al.

3 The BIT Flying Robot The BIT Flying Robot is aimed to monitor the internal environment of a Space Orbital Station. In particular, when an accident occurs, to ensure the safety of astronauts, a flying robot can enter the accident site instead of an astronaut to collect data and real-time images. Figure 2 shows a scheme for the structure design of BIT Flying Robot as for the conceptual design in Fig. 1 b). A command issued by the control unit either from internal strategy or astronaut’s indication is transmitted to the BIT Flying Robot through the communication module. The video module receives the command in coordination with the sensor module and driver status to acquire information. Then the data are transmitted to the motion planning for a proper motion planning unit that gives commands to the attitude and thrust modules to regulate the motion of the BIT Flying Robot. As indicated in Fig. 2, all the modules interact with the other modules for an integrated autonomous functioning.

Control Unit

communication module BIT Flying Robot

Driver status

Video module

Sensor module

communication module

Motion planning

Attitude module

Thrust module

Fig. 2. A scheme for the structure design of BIT Flying Robot as in Fig. 1 b).

Design and Testing of BIT Flying Robot

71

Figure 3 a) shows a CAD design with the six modules of BIT flying robot as for propulsion, battery, attitude, sensor, motion planning, and video systems whereas Fig. 3 b) gives an overview of its assembly in a built prototype. The battery modules and sensor modules of the flying robot are placed under the cover shell of the spherical volume. The attitude module is located inside the robot. The outer diameter of BIT flying robot is designed of 90 mm. The diameter of propellers is 7.0 mm. Thanks to a suitable small video system, the diameter of the board for video module is 36 mm. The diameter of the inside motion planning board is 30 mm. The total weight of the prototype in Fig. 3 b) is 0.27 kg as made of 3D printed plastic parts. Figure 3 c) shows the connection design with power bus supply and bus communication among the modules to limit the number of cables inside the spherical volume. Sensors in the prototype of Fig. 3 b) are limited to webcam, although room is available for more instrumentation in future developments.

.

a)

b) Battery Power cable

Power cable Communication model

Power cable

Motion planning IMU model

Video model Data cable

Power cable

Power cable

Thrust model Data cable

Data cable

Attitude model Data cable

Sensor model

c) Fig. 3. A design of BIT Flying Robot in Fig. 2: a) a CAD exploded view; b) the overall assembly in built prototype, c) the cable connections

72

Y. Liu et al.

4 Performance Characteristics Figure 4 shows main features of the designed operations of BIT Flying Robot. Figure 4 a) shows the environment detection through the video module and sensor module to monitor and check the environment of a space orbital station. When any failure occurs in the space station, a flying robot can detect and give feedbacks to astronauts. Figure 4 b) shows the module quick change of any parts of the BIT Flying Robot even with different sensor modules according to different tasks. For example, when the air quality needs to be detected, it is possible to simply remove the other sensor modules and to place the air quality sensor on it. Figure 4 c) shows the motion of the BIT Flying Robot using the attitude module to adjust its attitude and the thrust module to move it as planned. The above main features are implemented in the built prpotoype in Fig. 3 b).

module

Module Change

a)

b)

Motion decoupling

c) Fig. 4. Main operations of BIT flying robot: a) Environmental detection and space station diagnosis; b) Quick change of module parts, c) Thrust and attitude adjustment decoupling

Design and Testing of BIT Flying Robot

Τz

73

θRY

z

Yaw

ωAY

ωAP

Aƫtude module

ωAR

Τy

x

θRP

y

Τx

θRR

Roll

Pitch

Fig. 5. The scheme of the peculiar propeller

The propeller system is designed as in Fig. 5 to provide the thrust of the BIT Flying Robot, and the attitude module controls the attitude adjustment. The attitude module consists of three flywheels, each of which corresponds to yaw, pitch and roll. According to the principle of conservation of angular momentum, when a flywheel rotates, the BIT Flying Robot also rotates; when a flywheel stops, and the BIT Flying Robot also stops. The module is installed with the axes at the center of the BIT Flying Robot body. Decoupling the two modules is aimed to simplify control method with the expression. ⎡⎡

⎤ ⎡ ⎤⎤ ⎤ ⎡ θ¨RR 0 0 θ¨RR 0 0 ω˙ AR 0 0 JA ⎣⎣ 0 ω˙ AP 0 ⎦ + ⎣ 0 θ¨RP 0 ⎦⎦ = (JR + JA )⎣ 0 θ¨RP 0 ⎦ 0 0 ω˙ AY 0 0 θ¨RY 0 0 θ¨RY

(1)

where J A and J R represent the moment of inertia of attitude module and BIT Flying Robot body, respectively; ωAR , ωAP and ωAY represent angular rate of attitude module in roll, pitch and yaw; θ RR , θ RP and θ RY represent attitude angle in roll, pitcand yaw. The BIT Flying Robot’s thrust can be given by ⎤ ⎡ ⎤ aRx 0 0 Tx 0 0 ⎣ 0 Ty 0 ⎦ = mR ⎣ 0 aRy 0 ⎦ 0 0 Tz 0 0 aRz ⎡

(2)

where mR represents the mass of BIT Flying Robot; T x , T y and T z represent thrusts in x, y and z; α Rx , α Ry and α Rz represent the acceleration of BIT Flying Robot in x, y and z. The wing propellers work in coordination, mainly along the same axis. The expressions (1) and (2) are used for a continuous control of the motion by using closed-loop PID control algorithm within the control unit.

74

Y. Liu et al.

5 Prototype and Tests In order to simulate the environment with microgravity, a balloon is used to balance the gravity of the BIT Flying Robot during lab experiments. Balloon of 92 cm diameter is filled with helium for a buoyancy of 29.5 N. By adjusting counterweights, the BIT Flying Robot is suspended in the air simulating the microgravity environment. Figure 6 shows examples of using the BIT Flying Robot prototype in narrow space and particular conditions as in a Space Orbital station. Figure 6 a) shows an example of lab tests with BIT Flying Robot moving along and near a wall. Since the diameter of the balloon is much larger than the diameter of the BIT Flying Robot, a low-height cardboard box is used to simulate a wall. The BIT Flying Robot moves with a prescribed controlled motion at 10 cm from the wall. Figure 6 b) shows BIT Flying Robot moving inside a channel that consists of a space of 1.2 m width between a wall and table. Due to the diameter of the balloon and the connecting cable’s length, the maximum flying height of the BIT Flying Robo is 2 m during tests. Referring to a test of Fig. 6 b), the BIT Flying Robot run successfully along a straight line at a prescribed height and it exhibited a good performance in both thrust and attitude adjustment with results like those in Fig. 7 in terms of yaw motion. Balloon Channel Counterweight

Balloon Wall Counterweight

BIT Flying Robot

Connect line

Connect line BIT Flying Robot

Control Unit

.

a)

b)

Fig. 6. La tests of the BIT Flying Robot: a) Moving along a wall, b) Moving inside a channel

Due to the decoupling action of attitude adjustment and thrust, the BIT Flying Robot can adjust its attitude using flywheels. In the space, the exhaust or suction of the air by propeller will cause the BIT Flying Robot to shake at the target position. The robot adjusts its attitude with flywheels without using propellers and therefore it can prevent the BIT Flying Robot from shaking at a target position, saving energy and simplifying control. In Fig. 7, the target yaw rotating speed is prescribed as 1.47 rad/s (10 deg/s) during a test of Fig. 6 a). The adjustment of the yaw flywheel can make the BIT Flying Robot to have good maneuverability. It can be noted from Eq. (1) that when a flywheel starts to rotate, the BIT Flying Robot starts to rotate; when a flywheel stops rotating, the

Design and Testing of BIT Flying Robot

75

BIT Flying Robot also stops rotating. During a yaw motion of the BIT Flying Robot the balloon thin connecting cable will suffer torsion that will be returned to the BIT Flying Robot making necessary a continuous yaw adjustment as reported in Fig. 7. However, since in space, there is only air viscous drag torque, so this experiment will increase the thin wire deformation torque in the BIT Flying Robot system, which will cause that when the flywheel stops, which the target is zero, but the BIT Flying Robot still rotates. The plot in Fig. 7 shows good match between the planned and the experienced motions.

Fig. 7. Yaw attitude adjustment during a test of flying along a wall in Fig. 6 a).

6 Conclusions The paper presents the design and operations of BIT flying robot with its peculiarities that are illustrated by design developments and tested characterization. The BIT flying robot is built with a prototype for the Chinese space orbital station with a small size of 90 mm, thanks to proper modules for motion control and monitoring purposes.

References 1. Dorais, G.A., Gawdiak, Y.: The personal satellite assistant: an internal spacecraft autonomous mobile monitor. In: Proceedings of 2003 IEEE Aerospace Conference, Big Sky, pp. 1–348 (2003) 2. Fong, T., Provencher, C., Micire, M.J., Morse, T., Park, E.: Smart SPHERES: a telerobotic freeflyer for intravehicular activities in space. In: Proceedings of Space AIA 2013 Confenrence, San Diego, paper 5338 (2013) 3. Mitani, S.G., Masayuki, K., Ryo, S., Yasushi, H., Keiji, S., Shuhei, T., Nobutaka, S.: Int-Ball: Crew-Supportive Autonomous Mobile Camera Robot on ISS/JEM, pp. 1–15 (2019) 4. Liu, J., Gao, Q., Liu, Z.: Attitude control for astronaut assisted robot in the space station. Int. J. Control Autom. Syst. 14(4), 1082–1095 (2016) 5. Gao, Q., Liu, J., Tian, T., Li, Y.: Free-flying dynamics and control of an astronaut assistant robot based on fuzzy sliding mode algorithm. Acta Astronautica, 138, 462–474 (2017) 6. Zhang, R., Wang, Z., Zhang, Y.: Astronaut visual tracking of flying assistant robot in space station based on deep learning and probabilistic model. Hindawi International Journal of Aerospace Engineering, 2018, Article ID 6357185, p. 17

Surgical Skill Analysis Based on the Way of Grasping Organs with Forceps in Dissection Procedure of Laparoscopic Surgery Koki Ebina1 , Takashige Abe2 , Shunsuke Komizunai1 , Teppei Tsujita3 , Kazuya Sase4 , Xiaoshuai Chen5 , Madoka Higuchi2 , Jun Furumido2 , Naoya Iwahara2 , Yo Kurashima2 , Nobuo Shinohara2 , and Atsushi Konno1(B) 1

2 3

Graduate school of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan [email protected] Graduate School of Medicine, Hokkaido University, Sapporo 060-8638, Japan Department of Mechanical Engineering, National Defense Academy of Japan, Yokosuka 239-8686, Japan 4 Department of Mechanical Engineering and Intelligent Systems, Tohoku Gakuin University, Tagajo 985-8537, Japan 5 Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

Abstract. This paper presents a surgical skill analysis based on the way of grasping organs with forceps in dissection procedure in a wet lab training for laparoscopic surgery. Forty-four medical doctors and students conducted lymphadenectomy training using laparoscopic surgical instruments and porcine cadaver organs. During the training, movement of surgical instrument, grasping frequency/force/position, path length, completion time and other indices were measured. Among the indices, this paper focuses on analyzing the relationship between surgical skill level and the way of grasping organs with forceps such as grasping frequency, force, and position. The subjects are classified into three groups (novice, intermediate and expert) by the number of experienced surgeries. The indices between the three groups are compared by the Kruskal– Wallis and Mann–Whitney U test. The results of analysis show that the ratio of non-grasping period to the total task time of novice surgeons is larger than that of experts, and the total number of grasping of novices is greater than that of experts. From the results, it is found that the novices frequently re-grasp organs during the dissection operation. There is a possibility that novices frequently fail to grasp the organ at an optimal point and hence repeat re-grasping. Keywords: Laparoscopic surgery training measurement · Skill evaluation

· Grasping force

This work was supported by JSPS Grant-in-Aid for Scientific Research (A)(JP18H 04102). c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 76–83, 2021. https://doi.org/10.1007/978-3-030-58380-4_10

Surgical Skill Analysis Based on the Way of Grasping Organs

1

77

Introduction

Laparoscopic surgery is an operation performed within the abdominal or pelvic cavities. In the surgery, surgeons perform operation using surgical instruments such as endoscope and forceps, which are inserted through trocar on the patients’ abdomen. Compared to laparotomy surgery with a large incision, laparoscopic surgery generates is less-invasive, and therefore the burden of patients is light. Because of the less invasiveness, post-operative pain and the risk of infectious diseases are small, and quick recovery and short hospital stay are expected. From these backgrounds, laparoscopic surgery has become popular method in abdominal surgery. However, there are many difficulties in the surgery, and advanced techniques are required for surgeons. The visual information is limited because surgeons must perform operation by referring the two-dimensional laparoscopic monitor which displays the laparoscopy view of abdominal cavity. In addition, surgeons must get use to specific surgical instruments whose distance from the handle to the tip is long, and also cope with the inconsistency of hand-eye coordination [1]. In particular, incorrect manipulation of surgical instruments may cause complications such as organ damage and bleeding [2], and therefore efficient laparoscopic surgical training for novice surgeons is important. To develop laparoscopic surgical skill for novice surgeons, there are many types of training methods such as dry lab, wet lab, and VR (virtual reality) surgery simulator. The dry lab uses human organ model made of plastic or rubber, and the wet lab uses animal cadaver organs. However, the skill levels have not been quantified well because the surgical skill levels have been evaluated based on the simple criteria such as task complication time, and education for novice surgeons relies on the guidance of expert surgeons. Therefore, a demand for objective surgical skill evaluation for skill transfer from experts to novices is increasing. Laparoscopic surgical procedures are classified into basic operations such as forceps handling and cooperative movement of both hands, and applied operations such as dissection and suturing. In the dissection procedure, surgeons peel off and remove tissues or organs with grasping forceps and scissors forceps, and in the suturing procedure, surgeons suture tissues with needle holders. In particular, operational force of the grasping forceps in the dissection tasks is important to prevent unintended organ damage and bleeding. Yoshida et al. developed a system to measure the force on the tip of forceps by using a 3DoF force sensor, and conducted measurement experiments on the wet lab training for dissection procedures [3]. C. Richards et al. developed a measurement system for operational force and torque of grasper using a 6DoF force/torque sensor, and measured these data in cholecystectomy and Nissen fundoplication in a porcine model [4]. E. P. Westebring-van der Putten et al. investigated how different forms of haptic feedback influence the surgeon’s ability to generate a safe laparoscopic grasp through grasping force measurement [5]. Kuwana et al. developed grasping forceps with a 3DoF MEMS tactile sensor and quantified the stress generated organs [6]. However, these systems do not measures grasping force of forceps in

78

K. Ebina et al.

wet lab training. Grasping position of a grasping forceps is also considered as a key index of skill difference between experts and novices. In a discussion with expert urologists, the authors made the hypothesis that expert surgeons tend to grasp tissues with the forceps at the tip of the gripper, whereas novices tend to grasp them with the whole surface of the gripper. Motivated by this background, the authors developed a laparoscopic surgery measurement system that measures grasping force/position of grasping forceps with strain gauges [7,8], and conducting the measurement experiments for 46 subjects. This paper describes the detail of the measurement experiments and presents the obtained results of the experiments. In addition, to analyze skill difference between experts and novice surgeons, average grasping force, average grasping position, maximum grasping force, average of derivative of grasping force, ratio of non-grasping period to the total task time, and total number of grasping operation are chosen as evaluation indices. The results of the Kruskal–Wallis and Mann–Whitney U test conducted to compare the skill differences are also reported.

2 2.1

Measurement Experiments Experimental Environment Setup

Lymphadenectomy training in a laparoscopic training box using porcine cadaver organs is chosen for surgical skill evaluation. In order to measure the movement of surgical instruments, grasping frequency/force/position, path length, and completion time, the measurement system developed in [7,8] is used. This paper focuses on the relationship between the skill level and the way of grasping organs with forceps. In the lymphadenectomy training, grasping force/position of the grasping forceps are recorded with the sampling rate of 30 Hz. Figure 1 shows the experimental environment. The details of the grasping forceps used in the measurement are shown in Fig. 2. 2.2

Subjects and Task

The experimental subjects consist of 39 surgeons (gastroenterological surgeons, gynecologists, urologist, and junior doctors) and 5 students of Hokkaido University School of Medicine. Since two urologists performed the experiment twice, 46 measurement data were obtained. The subjects performed lymphadenectomy in which lymph nodes are dissected from the aorta using grasping forceps (KARL STORZ non-windowed grasping forceps, insert: K33310DY, sheath: K33300, handle: K33131), scissors forceps (OLYMPUS scissors forceps, scissors unit: A64810, sheath: A60800, handle: A60201), and clip appliers (Weck Hem-o-lok Polymer Locking Ligation System Applier size L). The aorta of porcine cadaver used in lymphadenectomy is shown in Fig. 3. The aorta of porcine cadaver is placed in a laparoscopic training box (see Fig. 1). Subjects conducted the training task using laparoscopic surgical instruments. The lymphadenectomy training was developed in [9] as an efficient training method for novice surgeons. In the training, an assistant operates the laparoscope camera.

Surgical Skill Analysis Based on the Way of Grasping Organs

79

Subject

Training box

Laparoscopic camera system

Laparoscope assistant

Fig. 1. Experimental environment

16 mm l (a) Overview

∑o

(b) Tip of gripper

Fig. 2. Grasping forceps

2.3

Experimental Results

Among the obtained measurement data of 46 cases, the data of a urologist are presented as a typical result in this section. The grasping force/position of the grasping forceps in the lymphadenectomy are plotted in Fig. 4. In this study, grasping position l is measured as a distance from the root of the gripper to the grasping point (see Fig. 2 (b)). In order to remove the noise included in the measurement data of the strain gauges, the Savitzky–Golay filter [10] is used. The polynomial order of the filter is set to 3, and the windows size of the filter is set to 15. Note that the grasping position data during the non-grasping period are unreliable and meaningless, therefore they are removed from Fig. 4 (b).

3 3.1

Skill Analysis Data and Analysis Indices

The relationship between the skill level and the chosen evaluation indices is studied using the measurement data of 43 cases. Among the 46 cases, the measurements in 3 cases are not successful. Therefore, the data of successful 43 cases

80

K. Ebina et al.

(a) Before the training

(b) After the training

Fig. 3. The aorta of porcine cadaver used in lymphadenectomy training 20 Grasping position (mm)

Grasping force (N)

20 15 10 5 0

0

200

400 600 Time (s)

800

1000

15 10 5 0

0

(a) Force

200

400 600 Time (s)

800

1000

(b) Position

Fig. 4. Grasping force and position

are used in the analysis. In the skill analysis, the subjects are classified into the following three groups according to their experience (number of surgeries Ns ): 1. Novices (N): (0 ≤ Ns ≤ 9) (9 subjects) 2. Intermediates (I): (10 ≤ Ns ≤ 49) (12 subjects) 3. Experts (E): (50 ≤ Ns ) (22 subjects) The novices consist of 2 junior doctors, 5 medical students, and 2 urologists. The experts consist of 9 gastroenterological surgeons, 3 gynecologists, and 10 urologists. All the intermediates are urologists. In this work, to evaluate the surgical skill of the subjects, the following indices are chosen as evaluation indices: (a) (b) (c) (d) (e) (f)

Average grasping force f¯ Average grasping position ¯l Maximum grasping force fmax Average of derivative of grasping force df Ratio of non-grasping period to the total task time Png The total number of grasping operation Ng

Surgical Skill Analysis Based on the Way of Grasping Organs (N) 9 8 7 6 5 4 3 2 1

(mm) 15 14 13 12 11 10 Experts

Intermediates Novices

Fig. 5. Average grasping force f¯ (N) 25

9

Experts

Intermediates Novices

Fig. 6. Average grasping position ¯ l (N/s) 1.75 1.5

20

1.25 1

15

0.75

10 5

81

0.5

Experts

Intermediates Novices

Fig. 7. Maximum grasping force fmax

0.25

Experts

Intermediates Novices

Fig. 8. Average of derivative of grasping force df

The data during the non-grasping periods are excluded in the calculation of average grasping force f¯, position ¯l, and average of derivative of grasping force df . 3.2

Results of Analysis

In order to investigate the difference between the three groups on the evaluation indices, Kruskal–Wallis test is used because the data normality cannot be assumed. Mann–Whitney U test is also used to investigate the difference of each pair of the groups. Figures 5–10 show the box plot of these indices of the three groups, and Table 1 shows the result of the tests. In the table, the indices E–N, E–I, and I–N are the p-values of the Mann–Whitney U test of the each pair of the groups, and the index E–I–N is the p-values of the Kruskal–Wallis test of the three groups. As shown in the Figs. 5–8, there are no significant differences in these indices. In particular, as discussed in the introduction, the authors made a hypothesis about the grasping position of the gripper of the grasping forceps. However, Fig. 6 shows that there are no significant differences in the grasping position between the three groups, and the analytical results of this study do not support the hypothesis. Alternatively, there are significant differences between the three groups in Png and Ng (see Figs. 9 and 10). As shown in Fig. 9, ratio

82

K. Ebina et al. p < 0.01

p < 0.01

(%)

200

p < 0.05

95 150

90

100

85

50

80

Experts

Intermediates Novices

Fig. 9. Ratio of non-grasping period to the total task time Png

0

Experts

Intermediates Novices

Fig. 10. The total number of grasping operation Ng

Table 1. Analytical results p-value

Median (Interquartile range) Experts (E)

E–I I–N

E–I–N

4.25 (3.44–5.34) 3.88 (3.13–4.76)

3.39 (2.85–3.86) ns

ns

ns

ns

13.0 (11.5–13.7) 12.9 (11.5–13.3)

13.1 (12.7–13.5) ns

ns

ns

ns

fmax 14.0 (10.4–15.4) 14.9 (12.9–17.8)

14.2 (11.9–17.3) ns

ns

ns

ns

df

0.88 (0.78–0.98) 0.86 (0.75–1.04)

1.00 (0.87–1.13) ns

ns

ns

ns

Png

93.0 (90.4–94.2) 90.5 (88.0–92.6)

86.9 (85.9–90.1) 0.0065 ns

ns

0.0234

134 (87–173)

0.0142 0.0006

f¯ ¯ l

Ng 31 (23.5–44.8) ns: Not significant

Intermediates (I) Novices (N)

40.5 (25–73.5)

E–N

0.0001 ns

Png of non-grasping period to the total task time of the novices is lower than that of experts, and therefore it is considered that the operation of the novices include the useless motion that does not contribute to the operation. In addition, as shown in Fig. 10, the total number Ng of grasping operation of the novices is greater than that of the experts. This result suggests that novices frequently re-grasp organs. There is a possibility that novices frequently fail to grasp the organ at an optimal point and hence repeat re-grasping.

4

Conclusion

This paper focuses on the relationship between surgical skill level and the way of grasping organs with forceps in dissection procedure of laparoscopic surgery. Among the evaluation indices, there are significant differences in the ratio of non-grasping period to the total task time and in the total number of grasping operation between the novices, intermediate, and expert groups. These results show that novices frequently re-grasp organs. It is expected that this result is reflected to the education of the laparoscopic surgery.

Surgical Skill Analysis Based on the Way of Grasping Organs

83

References 1. Wentink, B.: Eye-hand coordination in laparoscopy an overview of experiments and supporting aids. Minimally Invasive Therapy Allied Technol. 10(3), 155–162 (2001) 2. Inoue, T., et al.: Complications of urologic laparoscopic surgery: a single institute experience of 1017 procedures. J. Endourol. 24(2), 253–260 (2010) 3. Yoshida, K., et al.: Analysis of laparoscopic dissection skill by instrument tip force measurement. Surgical Endoscopy 48(27), 2193–2200 (2013) 4. Richards, C., et al.: Skills evaluation in minimally invasive surgery using force/torque signatures. Surgical Endoscopy 14(9), 791–798 (2000) 5. Westebring-van der Putten, E.P., et al.: Effect of laparoscopic grasper force transmission ratio on grasp control. Surgical Endoscopy, 23(4), 818–824 (2009) 6. Kuwana, K., et al.: A Grasping forceps with a triaxial MEMS tactile sensor for quantification of stresses on organs. In: Proceedings of the 35th Annual International Conference of the IEEE EMBS, pp. 4490–4493 (2013) 7. Ebina, K., et al.: A measurement system for skill evaluation of laparoscopic surgical procedures. In: Proceedings of the SICE Annual Conference, pp. 1099–1106 (2019) 8. Ebina, K., et al.: Development and validation of a measurement system for laparoscopic surgical procedures. SICE J. Control, Measurement, Syst. Integrat. 13(4), 191–200 (2020) 9. Higuchi, M., et al.: Development of a new simulation training model using swinetissue for laparoscopic dissection and suturing. In: Proceedings of the World congress of endourology, UP3-36 (2018) 10. Savitzky, A., Golay, M.J.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 36(8), 1627–1639 (1964)

Mechanism and Control of Powered Prosthesis with Bi-articular Muscle-Type Hydraulic Bilateral Servo Actuator Takanori Higashihara1,2(B) , Toru Oshima1 , Takumi Tamamoto1 , Kengo Ohnishi3 , Ken’ichi Koyanagi1 , and Yukio Saito4 1 Toyama Prefectural University, Toyama, Japan

[email protected] 2 Takamatsu Prosthetic and Orthotic Mfg., Ltd., Ehime, Japan 3 Tokyo Denki University, Saitama, Japan 4 Shibaura Institute of Technology, Saitama, Japan

Abstract. For person with bilateral shoulder disarticulation, powered prosthesis is highly needed. In late years, some experimental trials using electrically multi articulated powered prostheses has been achieved. However, these electrically powered prostheses are not so practical and hold unsolved problems such as its mechanism and control, fundamentally. To improve the functions, in this study, hydraulic-electric powered system were developed for a wearable prosthesis with 5 degrees of freedom. The wearable prosthesis can be supplied by the remaining functions of the human trunk to retain full workspace. In addition, this prosthesis can be used as a desktop mounted robot arm using an external attachment. This prosthesis have hydraulic bilateral servo actuator which combines the controllability of electric motor control and the high output of hydraulics. The joint mechanism is highly rigid mechanism without the use of speed reduction gear mechanism. A bi-articular muscle-type hydraulic bilateral servo actuator is used on the upper arm. Simultaneous and independent drive of the shoulder and elbow are possible with the one actuator. The mechanism and control of its upper arm with the bi-articular muscle-type hydraulic bilateral servo actuator are described in this study, in addition, sensor system using hydraulic pressure sensors which detect and avoid disturbance force is considered in this study for the safety control. Keywords: Shoulder disarticulation prosthesis · Bi-articular muscle-type actuator · Hydraulic drive · Safety control

1 Introduction Electrically powered prosthesis, which combine functionality with humanlike appearance, have become the standard prescription for below elbow amputees in European and North American countries in recent years. Electrically powered Shoulder prosthesis have been extremely difficult to be used practically. Uni/bilateral forequarter amputation or shoulder disarticulation is challenged of suitably adjusting the degrees of freedom, © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 84–91, 2021. https://doi.org/10.1007/978-3-030-58380-4_11

Mechanism and Control of Powered Prosthesis

85

mechanisms, and controllability, to fit the need to independently driving multiple joints in the shoulder, elbow, wrist, hand, and digits. However, with Kuiken’s surgical procedure, Targeted Muscle Reinnervation (TMR), to transplant the nerves of the remaining arm into the pectoral muscle and reinnervate the peripheral nerves have reduced the difficulty of gaining control signals for powered prosthesis. Concurrently, TMR amputees successfully operated DEKA Research & Development Corporation’s arm and Johns Hopkins APL arm with 25 degrees of freedom [1]. In Japan, in the latter half of the 1970s, Funakubo et al. at the University of Tokyo [2] and Kato et al. at Waseda University [3] developed electric-powered shoulder prostheses with six to seven degrees of freedom based on motion analyses of the human arm. In the latter half of the 1980s, Saito and Higashihara et al. at Tokyo Denki University [4] developed a highly rigid electrically powered shoulder prosthesis with a multi-joint space linkage shoulder mechanism with six degrees of freedom [5]. Developers of electrically powered shoulder prostheses have been focused on equipping the equivalent number of electric motors and gear reducers as the required degrees of freedom, with the ultimate goal of reproducing the equivalent freedom of motion as the human arm. However, such mechanisms are intolerable even in terms of weight alone, for amputees with disarticulation of both shoulders to wear the powered prosthesis in their daily lives. As a solution, a hybrid hydraulic-electric powered system for upper limb prosthesis is proposed. In using by wearable mode, the arm has 5 degrees of freedom and supplemented with advantage of the remaining capabilities of the trunk of the body. In using by desktop mode, the arm is supplemented with an external attachment with 2 additional degrees of freedom. This powered prosthesis installs hydraulic bilateral servo actuator that combines the controllability of electric motor control with the high output of hydraulic pressure and features a highly rigid joint mechanism without the use of speed reduction gear mechanism. Its bi-articular actuator properties enable simultaneous and independent driving of the shoulder and elbow. The forearm has a rotary-type actuator to achieve a highly rigid mechanism of 3 degrees of freedom. The electric hand allows 5 degrees of freedom, with each digit driven by a DC electric motor. And, as one of the safety measures at the time of contact with an object, three hydraulic pressure sensors mounted on the bi-articular muscle-type actuator for contact force detection.

2 Basic Mechanisms of Powered Prosthesis Figure 1 shows the powered prosthesis developed by in this research group equipped with a bi- articular muscle-type hydraulic bilateral servo actuator, and Fig. 2 shows the structure of this powered prosthesis with 5 degrees of freedom [6, 7]. A bi-articular muscle-type hydraulic bilateral servo actuator is used on the upper arm and simultaneous and independent driving of the shoulder flexion/extension θ 2 and elbow flexion/extension θ 3 are allowed. The forearm is equipped with rotary-type hydraulic actuators for forearm pronation/supination θ 4 , wrist dorsal/volar flexion θ 5 , and wrist radial/ulnar flexion θ 6 . In addition, the hand is equipped with 5 DC electric motors on each digit. This powered prosthesis does not have the mechanisms for shoulder adduction and abduction and inner and outer rotation of the upper arm that are found in previous electrically

86

T. Higashihara et al.

powered shoulder prosthesis. As shown in Fig. 3, the arm can also be used in its desktop form, where it is supplemented with an external attachment with 2 additional degrees of freedom corresponding to the shoulder, to consist 7 degrees of freedom arm.

Fig. 1. Picture of hydraulic driven prosthesis

Wearable

Fig. 2. Structure of hydraulic driven

Desktop

Circumntation of trunk Lateral bending of trunk

Fig. 3. Using by wearable mode/desktop mode

Fig. 4. Degrees of freedom of the trunk

One of the major challenges for previous electrically powered shoulder prosthesis are the implementation of mechanisms for shoulder adduction/abduction and inner/outer rotation of the upper arm. Our powered shoulder prosthesis does not have these mechanisms, yet it fully compensates this lack of degrees of freedom by actively utilizing the circumnutation and lateral rotation of the trunk, which are within the user’s residual function (Fig. 4). This motion of the trunk gives the prosthesis significant freedom of motion, eliminating the need for the shoulder based degrees of freedom. Table 1 shows the configuration of the powered prosthesis. Hydraulic actuator is implemented to the external attachment and arm actuators for the following reasons, 1) easy to achieve high output at minimal actuator size, 2) able to attach hydraulic actuator

Mechanism and Control of Powered Prosthesis

87

directly to the mechanism without speed reduction gear, 3) antagonistic actuation by the bilateral servo enables to maintain position even at power supply fails, 4) no other direct acting actuators superior to hydraulic cylinders. Table 1. Configuration of the powered prosthesis

Component

External attachment Arm unit

Hand unit

Articulated configuration of joints

Range of motion (deg.)

Shoulder

Inner / Outer rotation θ0 Adduction / Abduction θ1

–90 to –90 to

90 90

Shoulder Elbow

Flexion / Extension θ2 Flexion / Extension θ3

–40 to 0 to

83 113

Forearm

Pronation / Supination θ4

–90 to

90

Wrist joint

Dorsal / Volar flexion θ5

–90 to

60

–10 to

30

Radial / Ulnar flexion θ6 5 DC servo motors mounted on each digit

3 Control of the Shoulder and Elbow Joints by the Bi-articular Muscle-Type HBSA To conduct cooperative and independent driving of flexion/extension of the shoulder and elbow joints (2 degrees of freedom), the upper arm arranged bi-articular muscle-type hydraulic bilateral servo actuator (henceforth “HBSA”) [6]. Figure 5 shows a schematic diagram of the bi-articular muscle-type HBSA. The bi-articular muscle-type HBSA features a master slave design that offers high power and smooth operation characteristic of hydraulic pressure in the 2 cylinders without a hydraulic power source. The piston of the 2 ports master cylinder is driven by a servo motor and a ball screw feed mechanism. The hydraulic pressure acts on the bi-articular muscle-type slave cylinder through two solenoid valves, driving the pistons on both ends of the slave cylinder. Opening and closing the solenoid valves switches either independent driving of piston and coaxial driving of both pistons. Highly accurate position and force control capabilities are viable by a linear potentiometer and hydraulic pressure sensors, while stable control is achieved under open loop control. In addition, antagonistic actuation of the HBSA offers stable control characteristics that are not only high output but also independent of the magnitude of the load. The actuator constitutes the skeletal structure of the prosthesis. The master portion can be freely located outside the prosthesis, while the slave portion within the prosthesis is an extremely light actuator consisting of only the cylinder. Table 2 shows the direction of motion of the two piston rods Sps and Spe in the slave cylinder caused by the opening and closing of the solenoid valves Sa and Sb, and the motion of the master piston rod MP. The up and down arrows in the table correspond to the direction of motion of the piston rods in Fig. 5. Figure 6 illustrates schematically 3

88

T. Higashihara et al.

Shoulder

Shoulder

Elbow

Fig. 5. Schematic diagram of the bi-articular muscle-type HBSA

Elbow

3

3

Fig. 6. Motion of the shoulder and elbow joints by the bi-articular muscle-type HBSA

types of motion of the upper arm under by controlling the position of the master cylinder and opening and closing the two solenoid valves. If the valves Sa and Sb are closed, the master and slave cylinders become antagonistic stop. ➀: If the valve Sa is open and the valve Sb is closed, the master piston rod Mp produces flexion and extension of the shoulder joint. ➁: If the valve Sa is closed and the valve Sb is open, the master piston rod Mp produces flexion and extension of the elbow joint. ➂: If the valves Sa and the Sb are open, the master piston rod Mp produces flexion and extension of the shoulder and elbow joints simultaneously. The bi-articular muscle-type HBSA is controlled by a Table 2. Opening and closing of valves and motion of slave piston Sa Closed

Sb Closed

MP Stop

Open

Closed

Closed

Open

Open

Open

⬇ ⬆ ⬇ ⬆ ⬇ ⬆

Sps × × ⬇ ⬆ × × ⬇ ⬆

Spe × × × × ⬆ ⬇ ⬆ ⬇

Movement of the joint axis θ2 and θ3 Antagonistic stop Shoulder θ2 Elbow θ3 Shoulder and Elbow θ2 and θ3 simultaneous

Sa, Sb: Solenoid valve MP: Operation of the piston rod on the master cylinder Sps: Movement of the shoulder piston rod on the slave cylinder Spe: Movement of the elbow piston rod on the slave cylinder × : Stop ⬆ : Up ⬇ : Down

Flection Extension Flection Extension Flection Extension

① ② ③

Mechanism and Control of Powered Prosthesis

89

symmetric bilateral control system that feeds back the positions of the master and slave piston rods using potentiometer. In addition, hydraulic pressure sensors are installed in the three oil chambers to detect and prevent problems, such as the prosthesis colliding with objects in the environment.

4 Disturbance Force Detection with Pressure Sensors Mounted on a Bi-articular Muscle-Type Actuator for Safety Control

100

1.0

Middle pressure

Elbow pressure

Elbow angle θ3

0.5

50

0

-50 0 Closed Open

0

Shoulder pressure

Shoulder angle θ2 10 Open Closed

20 Open Closed

30 Closed Open

Pressure (MPa)

Shoulder & Elbow joint angle (deg.)

Safety of the powered prosthesis is one of the important problems. If the powered prosthesis in operation comes into contact with the user, other person or objects, there is a possibility of a serious damage. In general articulated robot, multi-axis force sensor is arranged in the wrist and force or contact disturbance is detected by the sensor. Multiaxis force sensor is very sensitive, but the demerits are its size and weight or the lack of strength. Therefore, as one of the safety measures at the time of contact with an object, three hydraulic pressure sensors mounted on the bi-articular muscle-type actuator for contact force detection. This sensor system can cancel the demerits of the force sensor mentioned above. Hydraulic pressure sensor can be detect the disturbance force act on the each joint of the whole arm of the prosthesis. This pressure sensor is high sensitive, very small, lightweight and robust compared with the force sensor. Figure 7 shows states of the hydraulic pressure sensor and the electromagnetic valve in the individual motion of the shoulder and the elbow of the bi-articular muscle-type HBSA. As shown in the Fig. 7, the pressure sensor value changes continuously during normal operation, but it is expected that the pressure sensor value will become discontinuous when disturbance force is applied to the arm. Therefore, the whole arm functions as a sensor that detects disturbance force.

Closed Open

40 Open Closed

Time (s) Valve Sa Valve Sb Mp

(Mp: Master position rod)

Fig. 7. Shoulder and elbow angles of bi-articular HBSA and output of three pressure sensors

Figure 8 shows the flexion/extension angle of the elbow and the output of the pressure sensor during individual operation in the bi-articular muscle-type actuator. The elbow

90

T. Higashihara et al.

100

1.0

Elbow angle θ3 Elbow pressure

Pressure (MPa)

Elbow joint angle (deg.)

was bent (0–90°) three times, ➀, ➁, ➂. The first and third motions did not accompany with the object contact, and a shock was applied during the second motion. Although the sensor values increase in accordance with the flexion of the elbow, it can be confirmed that the sensor value rapidly increases at a specific timing. This is the timing when a disturbance force was applied.

0.5

50

Middle pressure 0

Disturbance force 20

10

40 Time (s) 0

30

Fig. 8. Elbow angle and pressure sensor outputs during individual motion of the elbow in a bi-articular muscle-type

Avoidance behavior

78 Elbow pressure

Elbow angle θ3

0.5

74

70

Middle pressure 0

1

2

3

4

5

Time (s)

Fig. 9. Elbow avoidance behavior of bi-articular muscular actuator

0

Pressure (MPa)

Elbow joint angle (deg.)

Figure 9 shows the elbow avoidance behavior of the bi-articular muscular actuator. The avoidance behavior in this experiment is defined as “to move in the opposite direction for 1 s after the pressure sensor detects the impact”. The shock is set so that when it rises 0.1 MPa momentarily, it shifts to avoidance behavior. In the experiment, a pressure sensor can detect a force of 10 gf. In Fig. 9, the motion starts at about 1.1 s, and the avoidance behavior program becomes effective from about 1.6 s. A disturbance force is detected at about 3.0 s, the arm takes an avoidance behavior.

Mechanism and Control of Powered Prosthesis

91

5 Conclusions Our powered prosthesis is developed based on different design concept from previous powered prostheses, we propose a hybrid hydraulic electric powered system for shoulder prosthesis. In its wearable form, the arm has 5 degrees of freedom and supplemented with the advantage of the remaining capabilities of the trunk of the body. In its desktop mode, the arm is supplemented with an external attachment with 2 additional degrees of freedom. The powered prosthesis uses a hydraulic bilateral servo actuator that combines the controllability of electric motor control and the high output of hydraulic pressure. Its biarticular muscle model enables simultaneous and independent driving of the shoulders and elbows. Safety of the powered prosthesis is one of the important problems. In this prosthesis, the hydraulic pressure sensors in the bi-articular muscle-type hydraulic bilateral servo actuator can be detect the disturbance force on the arm from all directions. Traditional electric shoulder prostheses have used existing technology to replicate the freedom of movement of the human upper limb. This approach have resulted in significant progress including lighter electric motors, lighter and stronger components, and faster control systems. However, the fundamental design concept has not changed. Authors believe it is time to reconsider the relationship between mechanisms, degrees of freedom, and safety system of prosthesis.

References 1. Kuiken, T.A., Li, G., Lock, B.A., et al.: Targeted muscle reinnervation for real-time myoelectric control of multifunction prosthesis. JAMA 301(6), 619–628 (2009). https://doi.org/10.1001/ jama.2009.2.116 2. Funakubo, A., Yamaguchi, T., Saito, Y.: Motion analysis of human arms. Biomechanism 3, 97–103 (1975). (in Japanese) 3. Kato, I., Okada, Y., Ikeda, K.: Development of a hydraulic shoulder prosthesis. Biomechanism 3, 115–121 (1975). (in Japanese) 4. Higashihara, T., Saito, Y.: Mechanisms and engineering evaluation of a shoulder electricpowered prosthesis (TDU ys-86). Biomechanism 10, 227–237 (1990). (in Japanese) 5. Higashihara, T.: Present and future of control of prosthesis. J. Jpn Soc. Prosthet. Limbs 14(1), 16–25 (1998). (in Japanese) 6. Sakai, T., Saito, Y., Umemura, A.: A laparoscopic surgery training system and its application to a surgical support robot. Preprints from the 13th Japan IFToMM Conference Symposium, pp. 107–112 (2007). (in Japanese) 7. Higashihara, T., Oshima, T., Tamamoto, T., Ohnishi, K., Koyanagi, K., Saito, Y.: Mechanism and control of powered prosthetic arm using hydraulic bilateral servo actuators. Abstract of the 17th World Congress of the International Society for Prosthetics and Orthotics, Poster-5.075 (2019)

Development of a Remote-Controlled Drone System by Using Only Eye Movements for Bedridden Patients Atsunori Kogawa1 , Moeko Onda1 , Yoshihiro Kai1(B) , Tetsuya Tanioka2 , Yuko Yasuhara2 , and Hirokazu Ito2 1 Tokai University, Hiratsuka, Japan

[email protected] 2 Tokushima University, Tokushima, Japan

Abstract. In recent years, the number of bedridden patients with limb disabilities such as amyotrophic lateral sclerosis (ALS), has increased due to an extended life span and an aging society. Human eye movements are not relatively attacked even if ALS develops. Focusing on this point in this paper, we proposed a new remotecontrolled drone system by using only the patient’s eye movements in order to maintain and improve the quality of life of bedridden patients, including ALS patients. Furthermore, we presented experimental results to verify the effectiveness of the drone system. The experiment was conducted for three medical staff members who can evaluate this system considering bedridden patients. From the experimental results, we clarified what should be improved in order for bedridden patients to use this drone system. Keywords: Robot · Drone · Control system · Eye-tracking device · Quality of life

1 Introduction In recent years, the number of bedridden patients who have difficulty moving their arms or legs, including amyotrophic lateral sclerosis (ALS) patients, has increased due to an extended life span and an aging society [1, 2]. They are having difficulty performing activities of daily living (ADL), such as difficulty seeing the scenery outside of their rooms by themselves [3]. Improving their quality of life (QOL) is needed. In order to maintain and improve the QOL of bedridden patients, various kinds of robot systems have been developed [4–7]. Patients’ electroencephalogram signals (EEG), electromyogram signals (EMG), breathing, or voice [8–11] are used to control the robot systems. However, when using EEG, EMG, or breathing, there are sanitary issues. For example, to use EEG or EMG, putting electrodes on the patients is needed. Also, to use their breath, inserting a device for measuring their breath in their mouth is needed. When using their voice, producing a sound is needed. However, some ALS patients have difficulty producing sounds [12]. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 92–99, 2021. https://doi.org/10.1007/978-3-030-58380-4_12

Development of a Remote-Controlled Drone System

93

On the other hand, human eye movements are not relatively attacked even when ALS develops [13]. Therefore, we consider that a robot system controlled by gaze is useful for bedridden patients, including ALS patients. Moreover, in the robot systems, wheeled robots or legged robots are mainly used [14]. However, the patients might feel stress when the robots cannot climb over obstacles. Hansen et al. have developed a drone system controlled by using an eye-tracking device and a keyboard [15]. The patients will not feel stress because the drone can fly over obstacles. However, the patients who have trouble using keyboards will have difficulties using this system. To solve these problems, Adachi et al. have proposed a drone system operated by only eye movements [16]. In this system, by using a unique control screen (including an eye-tracking device) considering microsaccade, etc., it is possible for the patients to operate the drone with only their gaze and see a scenery reflected through the drone camera. However, in this system, they cannot see the scenery of a remote place (for example, overseas) and talk with people in the remote place. Also, the drone (Bebop Drone, Parrot S.A.) used in this system does not have an obstacle detection function. Therefore, it has the risk of collision with obstacles such as walls during flight. In this paper, we propose a remote-controlled drone system using the Internet, Skype, an eye-tracking device, and a drone with an obstacle detection function to further improve the QOL of bedridden patients. By using this system, bedridden patients will be able to operate the drone in a remote place with only their eyes. Furthermore, by using this system, they will be able to enjoy the scenery of a remote place and talk with people in the remote place. Moreover, the drone in this system will be prevented from colliding with obstacles by using the obstacle detection function. This paper is organized as follows. First, we explain the outline and the control method of this system. Next, we present experimental results to verify the effectiveness of the drone system. Finally, from the experimental results, we clarify what should be improved in order for bedridden patients to use this drone system.

2 System 2.1 Overview Figure 1 shows the drone system proposed in this paper. This system consists of a drone (SPARK, DJI), a control screen, an eye-tracking device (Tobii Eye Tracker 4C, Tobii), Computer A, Computer B, the Internet, Skype, a joystick control device, a controller (Spark Remote Controller, DJI), and a smartphone (iPhone XS, Apple). The drone has an obstacle detection function to prevent from colliding against obstacles. The control flow is as follows: (i) The eye-tracking device attached to the control screen detects the gaze position of the patient looking at the control screen. (ii) The gaze position data is sent from Computer A to Computer B via the Internet. (iii) The control signal data based on the gaze position data is sent from Computer B to the joystick control device, then the drone is operated by controlling the joysticks of the controller. (iv) The camera videos of the drone are displayed on the control screen via the controller, the smartphone operation application (DJI GO 4, DJI), the smartphone, Computer B,

94

A. Kogawa et al.

Drone

Control screen Patient

Eye-tracking device

Smartphone

Controller

Skype Internet

Computer A

Computer B

Joystick control device

Gaze position data Drone’s camera images Fig. 1. Proposed drone system.

Skype, and Computer A. Thus, the patient can control the drone and see the scenery of a remote place. 2.2 Control Method Figure 2 shows the control screen and the drone’s coordinate system. The control screen was made by adding an area to the control screen devised by Adachi et al. [16] in order to control the drone taking off. For the control screen, a 23-in. monitor (286.3 mm by 508.3 mm) was used. The control screen devised by Adachi et al. was designed considering field of vision, microsaccade, and eye strain. Therefore, based on Ref. [16], we set the distance between the control screen and the patient’s eyes at 550 mm and r = 19.2 mm. Furthermore, we set a = 38.4 mm and b = 96.0 mm in Fig. 2. As shown in Fig. 2, this control screen is divided into 7 areas. Area 1 is a circle at the center of the control screen and does not include Area 2. Area 2 is a circle below the center of the control screen and inside Area 1. Lines A and B are tangent lines in the Z direction on Area 1’s circle. Area 3 is on the right side of Line A, and Area 4 is on the left side of Line B. Area 5 is above Area 1, and Area 6 is below Area 1. Area 7 is above Area 5. The control method of the drone is as follows: 1. While the patient looks at Area 1, the drone hovers. 2. While the patient looks at Area 2, the drone moves forward (If the drone detects an obstacle within 1.5 m forward, then it stops moving forward and hovers.) 3. While the patient looks at Area 3, the drone rotates right. 4. While the patient looks at Area 4, the drone rotates left. 5. While the patient looks at Area 5, the drone ascends. 6. While the patient looks at Area 6, the drone descends. Also, if the patient continues to look at Area 6 until the drone’s fuselage touches the ground, it lands.

Development of a Remote-Controlled Drone System

95

508.3 mm b

Area 7 +Z

a

(Take off)

r

286.3 mm

Line B

Area 4 (Rotate left)

Area 5 +Z (Up) Area 1 (Hover) r

Line A 5r

Area 3 + (Rotate right)

+

+Z Forward

Area 6 -Z (Down, land) Z

Area 2 +X (Forward)

Eye-tracking device X Control screen

Drone’s coordinate system

Fig. 2. Control screen and drone’s coordinate system.

7. If the patient looks at Area 7 for 1.8 s, the drone takes off to a height of 1 m. If the patient continues to look at a target which he/she wants to see, the drone moves so that the target moves toward Area 1, and then finally he/she can see the target calmly in Area 1. The patient is instructed to basically look at the center of each area. When the patient mistakenly has looked at an unintended area, he/she can operate the drone as intended by looking at the intended area again.

3 Experiment 3.1 Subjects The healthy subjects who participated in this experiment were two male (one with glasses, one with contact lenses) and one female (with contact lenses) medical staff members (nurses, average age: 45.7 years old). All three subjects operated this system for the first time in this experiment. They had enough clinical experiences to evaluate this system considering bedridden patients. 3.2 Experimental Procedure In this experiment, we prepared a room for the subjects (Room A) at Tokushima University and a room for an experimenter (Room B) at Tokai University, and the subjects controlled the drone along a flight course set in Room B from Room A via the Internet. They were able to talk to the experimenter in Room B using a speaker, a microphone, and Skype.

96

A. Kogawa et al.

First, we explained the control method and the flight course to the subjects, then calibrated the eye-tracking device for each subject and started this experiment. After the experiment, they answered a questionnaire about the evaluation of operability. 3.3 Flight Course Figure 3 shows the flight course in this experiment. The letters showed on Target A and Target B are randomly changed for each subject. Target C attached to a wall at the front center is a red vertical line that determines the direction to the goal. The subjects controlled the drone in the following procedure: (i) The subject looks at Area 7, then the drone takes off and raises to a height of 1 m. (ii) The subject rotates the drone 90° to the right. If the subject recognizes the letter on Target A, the subject says the letter. (iii) The subject rotates the drone 180° to the left. If the subject recognizes the letter on Target B, the subject says the letter. (iv) The subject rotates the drone 90° to the right and looks at Target C. (v) The subject advances the drone until the obstacle detection function is activated. (vi) The subject performs landing operation. If the flight is failed, the experiment is repeated until successful. 3.4 Evaluation Method Table 1 shows the questionnaire evaluation results. The subjects answered whether each of the questions was true or not. In order to clarify what should be improved in this system, we discussed with the subjects after the experiment. 3.5 Results and Discussion As a result of this experiment, all three subjects were successful at their first flight. The average flight time was 62 s. The standard deviation was 17 s. The right side of Table 1 shows the number of respondents who answered true. One of the three subjects answered “yes” to “Is there any freezing of the video during operation?” and to “Is there a delay between the video and drone operation compared to my own gaze?” However, since all the subjects were able to successfully fly in the first flight, it can be said that this control system can fly the drone without any problem if the video is disturbed as much as in this experiment. Additionally, the obstacle detection function was activated when the drone approached the wall during flight. After this experiment, the subjects stated that they were able to operate with peace of mind due to the obstacle detection function.

Development of a Remote-Controlled Drone System

1.5m Target A 1m

1.5m

1.2m

1.2m 1.2m

1.5m

1.2m 1.5m

1.5m

1m

1m

Target C

5. Move forward

6. Land Goal area

Goal area

Goal area 1m

Target B

1.2m

Wall

Target C

4. Rotate right

1m

3. Rotate left

1.5m

1m

1.2m

1.5m

1.2m

2. Rotate right

Start

1m

1m

1.5m

1.5m

1.5m

1. Take off

1.2m

1.2m

1.5m

1.2m

1.2m

1.5m

1.2m

97

1m

1m

1m

1m

Fig. 3. Flight course.

4 What Should Be Improved in This System? From the results of the questionnaire and discussion after the experiment, we clarified what should be improved in order for bedridden patients to use this system. First, it was discussed that patients might forget what each area on the control screen is. As a result, in order for patients to understand each area easily, it was clarified that the word or the letter representing what each area is should be written in the area. For example, “Rotate Right” or “R” should be written in Area 3, “Up” or “U” should be written in Area 5, and so on. We determined that this is useful for preventing from forgetting how to control the drone while operating. Second, it was discussed that patients may feel eye strain during the operation. As a result of the discussion, it was clarified that the drone system should have functions where the drone hovers when the patient using this system closes his/her eyes during operation, and where the drone lands if he/she keeps his/her eyes closed for 10 s. Third, it was discussed that the control method should be explained more easily in order for patients to understand. As a result, it was clarified that explaining on how to use the system by using a promotional video, etc. is important.

98

A. Kogawa et al. Table 1. Questionnaire results.

Questions

Number of respondents

1

Is there any blurriness in the video during operation?

0

2

Is there any distortion in the video during operation?

0

3

Is there any vanishing effect in the video during operation?

0

4

Is there any noisiness in the video during operation?

0

5

Is there any freezing of the video during operation?

1

6

Is there a delay between the video and drone operation?

1

7

Is there a time when the drone is uncontrollable during operation?

0

5 Conclusion In this paper, in order to maintain and improve the QOL of bedridden patients, including ALS patients, we have developed a remote-controlled drone system operated by using their gaze. We conducted an experiment in which three medical staff members used the developed drone system and evaluated the effectiveness. As a result, all three medical staff members succeeded in the first flight. In addition, considering that bedridden patients will use this system in the future, we clarified what should be improved in this system. In our future work, we will improve the drone system based on the results discussed in Sect. 4 and verify the effectiveness of the improved system by experiments. In addition, it is also necessary to conduct experiments for verifying whether the drone is controlled as bedridden patients intend. Moreover, we also need to consider or improve the following for practical use: (i) the drone’s flight range, (ii) the noise created by the drone, (iii) the regulations that exist in many places regarding the use of drones, (iv) safety measures in case of network errors, etc. This research was conducted through the ethical review and the research approval of Tokai University (Approval number: 20037). Acknowledgement. This research was partially supported by the Pfizer Health Research Foundation.

References 1. Mizoguchi, K., Chida, K., Sonoda, Y., Mori, T., Funakawa, I., Yorimoto, K., Uekawa, K., Yuasa, T.: Amyotrophic Lateral Sclerosis (ALS): a new framework and community healthcare

Development of a Remote-Controlled Drone System

2. 3.

4. 5. 6. 7.

8.

9.

10.

11. 12. 13. 14. 15.

16.

99

support -Expectations for the National Hospital Organization-. IRYO Jpn. J. Nat. Med. Serv. 61(7), 490–500 (2007). (in Japanese) Cabinet Office: Reiwa 1st Edition White Paper on Ageing Society. Nikkei Printing Inc., Tokyo (2019). (in Japanese) Kato, H., Hashimoto, R., Ogawa, T., Tagawa, A.: Activity report on the occasion of the 10th anniversary of the center for intractable neurological diseases. J. Int. Univ. Health Welfare 19(1), 16–23 (2014). (in Japanese) Yamamoto, M.: An input interface device using breath pressure for severely handicapped. J. Robot. Soc. Jpn 28(9), 1079–1081 (2010). (in Japanese) Fukuda, O., Tsuji, T., Kaneko, M.: A human supporting manipulator based on manual control using EMG signals. J. Robot. Soc. Jpn 18(3), 387–394 (2000). (in Japanese) Komeda, T., Funakubo, H.: Development of an environmental system for bedridden patients. J. Life Supp. Technol. 1(2), 61–67 (1986). (in Japanese) Iwasaki, M., Maruhashi, K.: Initiatives for social integration of bedridden patients: the challenge with the iPad. In: Proceedings of Joint Congress of Physical Therapist and Occupational Therapist in Kyushu, p. 297 (2017). (in Japanese) Naveen, R.S., Julian, A.: Brain computing interface for wheel chair control. In: Proceedings of the Fourth International Conference on Computing, Communications and Networking Technologies (ICCCNT), pp. 1–5 (2013) Wang, W., Zhang, Z., Suga, Y., Iwata, H., Sugano, S.: Intuitive operation of a wheelchair mounted robotic arm for the upper limb disabled: the mouth-only approach. In: Proceedings of the 2012 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 1733– 1740 (2012) Nagy, G., Várkonyi-Kóczy, A.R., Tóth, J.: An anytime voice controlled ambient assisted living system for motion disabled persons. In: Proceedings of the 2015 IEEE International Symposium on Medical Measurements and Applications (MeMeA), pp. 163–168 (2015) Inoue, Y., Kai, Y., Tanioka, T.: Development of a partner dog robot. Bull. Kochi Univ. Technol. 1(1), 52–56 (2004). (in Japanese) National Hospital Organization, Utano Hospital: Neuromuscular Intractable Disease Nursing Manual. Nissoken Shuppan, Tokyo (2004). (in Japanese) Ogawa, A.: Cranial Nervous Disease: Think from Pathophysiology and Nursing Point Q & A 200. Medicus Shuppan Publishers Co., Ltd., Osaka (2004). (in Japanese) Yamashita, A., Asama, H., Arai, T., Ota, J., Kaneko, T.: A survey on trends of mobile robot mechanisms. J. Robot. Soc. Jpn 21(3), 282–292 (2003). (in Japanese) Hansen, J.P., Alapetite, A., MacKenzie, I.S., Møllenbach, E.: The use of gaze to control drones. In: Proceedings of the Symposium on Eye Tracking Research and Applications (ETRA 2014), pp. 27–34 (2014) Adachi, Y., Kai, Y., Yuyama, T., Hayama, J.: A UAV system using an eye-tracking device for patients with limb disabilities: design of its control screen. In: Proceedings of the SICE Annual Conference 2018, pp. 854–859 (2018)

Development of a Climbing-Robot for Spruce Pruning: Preliminary Design and First Results Giovanni Carabin(B) , Davide Emanuelli, Raimondo Gallo, Fabrizio Mazzetto, and Renato Vidoni FAST, UNIBZ, Bolzano, Italy {giovanni.carabin,raimondo.gallo,fabrizio.mazzetto, renato.vidoni}@unibz.it, [email protected]

Abstract. The aim of the project in which this work was conceived is the design of a climbing robot for pruning spruce trees without damaging the bark. According to the requirements and pruning methods for avoiding the flaws of the finished timber, guidelines and constraints for the mechatronic design of a new climbing robot for spruce-pruning activities were deduced. A 3-wheeled-driven system able to cope with the tree bark thanks to compression springs is chosen for the first functional design; a force-balance analysis is performed for understanding the climbing and equilibrium requirements also considering the additional weight of a pruning system installed on a cart moving on a guide placed on the robot main structure. The quasi-static model, the mechanical design and the related electronic and driver systems are here presented. The preliminary scaled mechatronic prototype is experimentally evaluated on poles with diameters in the range of the young spruces, i.e. 100–200 mm, in its main climbing features. These results will serve as basis for the further development of the system. Keywords: Climbing-robot Robotics

1

· Spruce-pruning · Mechatronic design ·

Introduction

A proper spruce-tree pruning is important for ecological health and preventing wildfire as well as for the commercial viability of a tree stand. For the former, when the trees that compose a forest stand increase in size, the branches in their crowns begin to touch each other creating a shield to the entrance of the solar light. As a consequence, all those branches which do not participate efficiently in the photosynthetic processes are excluded from the nutrient exchange process, causing their complete dry out and death. In this condition, the branches may broke due to external interactions, i.e. “natural pruning”. As a result, dried wood fragments of different lengths can remain attached to the trunk lowering the quality of future timber as well as possibly fueling wild-fires [1]. For the c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 100–108, 2021. https://doi.org/10.1007/978-3-030-58380-4_13

Development of a Climbing-Robot for Spruce Pruning

101

latter, pruning of spruce trees in a proper way allows the increase of the quality of the timber and, on the contrary, an incorrect pruning can expose the tree to fungi invasion or, in the worst case, can permanently damage the tree [1–3]. This operation is called “green pruning”. It is usually performed in profitable stands by operators that remove branches using ladders, crampons (metal climbing cleats), manual saws or light chainsaws [4]. Since the 1970’s, the forest engineering community has had an enormous interest in the development of automatic machines that can climb trees and prune branches with a low number of cutting flaws. However, to date and to the authors’ knowledge, no effective pruning robots have been developed and commercialized. Few examples of automatic machines for pruning trees can be found (e.g. Advaligno Patas [5] and the Tree-Monkey [6]) and, even if these machines are fast, they are heavy, do not allow a controlled pruning and, thus, they are not suitable for the chosen activity. In this context, the usage of a robot to climb, sense the branches and prune trees can be considered to be a relatively young and challenging research area. Accordingly, there are few references in the technical literature concerning such devices. On the contrary, many attempts have been proposed for climbing walls and trees with robots by implementing a continuous or step-by-step motion: – Continuous motion: gripping mechanisms inspired by lumberjacks; passive, preloaded spring gripping mechanisms; active gripping mechanisms; parallel robot climbing kinematics; snake inspired robots (e.g. [7–10]). – Step-by-step motion: body extension and contraction; legged, insect-inspired robots; inchworm-like robots (e.g. [11–15]). The special requirements of a spruce-pruning robot require appropriate technical solutions. The main/minimum requirements can be summarized in: – the final design should overcome the main limitations in size and weight of the current commercial systems (e.g. be operated by only one worker); – a good adherence with the bark has to be guaranteed without using spikes and avoiding the creation of wounds along the trunk (the bark of spruce is thin and tender during juvenile age, easy to be damaged). – the robot should be designed to (auto-)adjust around the trunk, ensuring the capability to operate with different diameters (usually from 12 to 20 cm); – the green pruning is compulsory in order to avoid dead knots; – the cut operation should be done at the right (inclination and) distance from the trunk, preserving the branch collar; – the system autonomy, if not tethered, has to allow a working day activity. In the following, the first functional and mechatronic design for a climbing robot that could be equipped with a pruning system and satisfy the needed requirements is presented and discussed. Moreover, a preliminary prototype and related experimental results are presented.

102

2

G. Carabin et al.

Modelling and Conceptual Design

The main concept and chosen solution for the climbing-robot is presented in Fig. 1. It is made of a circular ring structure connected through revolute joints to three evenly spaced legs equipped at their extremities with an actuated-wheel system; each leg is thought in a way that elastic springs, i.e. passive systems, connecting the main frame with each leg allow a proper contact force between the tree-trunk and each wheel. The circular ring main frame is made of three modules connected via hinges. It can then be opened and closed to embrace the trunk of the tree in a fast and simple way (i.e. from around 100 mm up to 200 mm of diameter). On its top plate/face, a guide that hosts a motor-driven cart to circumnavigate the trunk is chosen as solution for the cutting system. Indeed a chainsaw or a cutting element can be mounted on the cart and moves around the circular ring to execute the pruning operation.

Symbol Description R r l M (d, ϑ) Fi μi Fi μ

Trunk radius Ring frame radius Arm length Robot weight CoG position i-th arm normal force i-th wheel friction force Wheel-bark static friction coefficient

Fig. 1. Conceptual model of the pruning-climbing robot.

Referring to Fig. 1, three normal forces F1 , F2 and F3 , generated by the action of the wheels against the bark, have to assure the right friction force (i.e. μ1 F1 , μ2 F2 and μ3 F3 , with μi the friction coefficient) to counteract the weight M . In this preliminary study the dynamic effects are neglected, since the robot is thought to reach low acceleration/deceleration values both in the climbing motion as well as in the pruning one. Moreover, the worst case scenarios are considered to study both the stability in a steady (pruning) condition and the minimum conditions for climbing. In particular, from the stability point of view, the worst case occurs when the arm-springs are in their minimum-load condition, i.e. smallest trunk diameter, and when the robot is perfectly aligned with the plane normal to the tree main axis. Indeed, in this condition, the three contact points are on the circle with the diameter of the trunk. If the robot starts tilting, the contact circle deforms into an ellipse and at least one spring arm results to be more loaded. Moreover, in the robot balanced configuration, the gravity force exerts the maximum torque around the x and y axis due to the longer lever arms. With reference to Fig. 1, the static equilibrium of the system can be expressed through the following system of equations:

Development of a Climbing-Robot for Spruce Pruning

⎧ F1 − 1/2F2 − 1/2F3 = 0 ⎪ ⎪ ⎪ √ √ ⎪ ⎪ 3 3 ⎪ ⎨ /2F2 − /2F3 = 0 μ1 F1 + μ2 F2 + μ3 F3 − M g = 0 ⎪ √ √ ⎪ ⎪ ⎪ μ2 F2 3/2R − μ3 F3 3/2R − M g d sin ϑ = 0 ⎪ ⎪ ⎩ − μ1 F1 R + μ2 F2 1/2R + μ3 F3 1/2R + M g d cos ϑ = 0

103

(1)

where d and ϑ identify the location of the CoG (center of gravity). From the first two equations it results that F1 = F2 = F3 = F and, thus, by solving for μ1 , μ2 and μ3 , it holds: √ R + 2 d cos ϑ M g R − d cos ϑ ± 3 d sin ϑ M g μ1 = μ2,3 = (2) 3R F 3R F The instability of the climbing platform happens when one or more wheels start sliding, i.e. the friction force is overcome. Thus when μ1 , μ2 or μ3 are greater than the static friction coefficient μ (e.g. μ = 0.9 for soft rubber – wood materials). Considering the worst critical stability case (i.e. minimum trunk radius R and ϑ = 0 + 2π/3), the Eq. (2) can be solved for different CoG distances d from the center and for different values of the ratio between the gravitational force M g and the normal force F that each wheel applies against the tree bark. By introducing a stability index (SI) of the climbing platform expressed as SI = 0 3 if at least one μi is zero (i.e. instability), or SI = i=1 (1 − |μi |/μ) in the other cases, a map like the one in Fig. 2a can be obtained.

90 60 30 0 -30 -60 -90 30

1

1.1 1

0.9

0.9

0.8

0.8

0.7

35

40

45

35

40

45

35

40

45

200

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

26

0.3

0.2

25

0.2

0.1

24

0.1

0

150 100 50 30

30

40

50

(a)

30

60

(b)

(c)

Fig. 2. (a) Map of the platform stability index SI versus d and the R = 0.05 m and ϑ = 0 + 23π ; (b) spring model, and (d) spring design.

M g/F

ratio with

104

G. Carabin et al.

It can be noted that the CoG distance from the center d has a limited effect on the stability as the CoG is inside the trunk circumference (i.e. d < Rmin = 50 mm). Within this evaluation, with a M g/F ratio lower than 0.8 the feasibility of the climbing operation is assured (i.e. assuming that d is always lower than 50 mm). This condition is guaranteed by the action of elastic elements that push each of the arms on the trunk generating the normal force F . As shown in Fig. 2b, the elastic torque is obtained by connecting a linear spring of elastic constant k with a disc of diameter Dd directly coupled with the arm. Neglecting the positive effect of the friction force, the momentum equi2 librium can be thus expressed as k D4d (ϕ − ϕ0 ) + Fk0 D2d = F l sin ϕ, where  r−R  and ϕ0 and Fk0 are the no-load angular position and the initial ϕ = acos l tension of the elastic element, respectively. The equation can be rearranged and solved to obtain the plots like the ones shown in Fig. 2c: the first plot shows the initial (angular) position ϕ0 of the spring, the  secondreports the (angular) elongation of the spring calculated as Δϕ = acos r−Rlmax − ϕ0 , and finally the third plot indicates the maximum force F that the wheel generates on the bark when the robot climbs the largest trunk.

3

Mechatronic Design and Prototyping

Fig. 3. Climbing robot prototype: Design (a) main assembly, (b) frame, (c) arm and (d) rail and cart; Prototype (e) main assembly, (f) application on a tree trunk, (g) detail of the arm spring and (h) robot parameters.

The evaluation of the stability model and of the conceptual design have been done by designing a preliminary version of the platform in Solidworks (Figs. 3a

Development of a Climbing-Robot for Spruce Pruning

105

and 3e) and by prototyping it using plexiglass and a laser cutter machine. This allowed us to realize a scaled system in terms of efforts and weight but implementing feasible geometrical dimensions and features. The circular ring frame, equipped with two hinges and a closure lever is visible in Fig. 3b: an easy and quick open and close around a tree trunk is guaranteed (Fig. 3f). The design of the three arms is reported in Fig. 3c: on one extremity the DC geared motors and the wheels are attached, on the other they are connected to the frame by means of revolute joints. The main dimensions and parameters of the designed and prototyped system are reported in Fig. 3h. Figure 3g shows the elastic elements introduced to create the proper wheel pressure against the trunk according to the equations presented in the previous section. Considering the parameters in Fig. 3h and the plots in Fig. 2c, several values of the spring parameters (i.e. k and Dd ) can be chosen and trade-off values of k = 5 N/mm and Dd = 40 mm (i.e assuming Fk0 = 0.03 k) to both allow a correct preload as well as to not to cause an excessive pressure on the bark have been selected. Figure 3d shows the circular rail with a double T section shape and the cart with four wheels where one is actuated and a tilting axle allows to follow the circular path. The cutting device is going to be fixed on the cart possibly through a passive mechanism assuring the right cutting distance of the branch from the trunk. The robot climbing motion is guaranteed by three Pololu 12 V 37D 1:131 geared DC motors equipped with 64 CPR encoders. They have a maximum torque of 1.8 nm and a free-run speed of 80 rpm: a theoretical climbing speed of 0.34 m/s (i.e. 80 mm of wheel diameter) can be achieved. A further smaller geared DC motor has been used to actuate the cart motion along the rail for the cutting operation. All the motors are driven by two Cytron MDD3A boards, a dual channel 3 A, 4 ÷ 16 V motor drive. A Teensy 4.0 board (ARM Cortex-M7 at 600 MHz, 1024 K RAM) has been selected as control system: its function is to receive the start command, that includes the climbing velocity of the robot (for now from an external PC) and drive properly the three wheels. In particular, three speed loop PI controllers running at 1 kHz have been implemented and tuned to assure the proper climbing operation. Speeds are measured from the encoder signals and an average base filtering has been considered to remove the noise. In this first prototype version the power is fed via an external source.

4

Results and Discussion

In order to evaluate the effectiveness of the prototype performing a climbing activity as well as the validity of the model, some static and dynamic tests have been made. For the former, the platform stability has been verified in presence of balanced and unbalanced loads, Fig. 4a). The robot has been tested when coping with plastic tubes with different diameters (i.e. 125, 140, 160 and 200 mm); extra weights have been added on the cart (e.g. variation of the CoG position) up to one of the wheels started to slide. In Fig. 4b the resulting map is shown for all the considered R and ϑ values. Note that, due to the symmetry of the system,

106

G. Carabin et al.

only three values of ϑ have been considered (i.e. 0, 30 and 60◦ ) and the results transferred properly to the other angles. Looking at the figure it can be seen how for small diameters, low values of unbalancing mass (i.e. 0.2 kg) could cause the robot instability. Anyhow, the tests have been performed using plastic tubes and the results are thus very precautionary (i.e. with a real trunk wheels would have a higher grip). A further test concerns the measure of the actual normal force F generated by each wheel on the bark surface. As visible in Fig. 4c, three load cells have been arranged at 120◦ on three linear guides to change the diameter and they have been equipped with plates on which the wheels apply the force. In Fig. 4d the linear trend of the force versus the trunk radius is shown: the variation of the force is very limited from the smaller radius to the larger, and smaller than the one predicted from the model (i.e. from 16 N to 24 N). This is probably due to the friction inside the system revolute joint, and the friction between the spring and the disc surface. Finally, preliminary dynamic tests with low climbing-speed have been performed on tubes of different diameters showing the effectiveness of the system and thus allowing to confirm the climbing and descending capabilities of the prototype.

Fig. 4. (a)(b) Results of the unbalanced load test: the map shows the maximum mass of the extra weight added on the cart on different angular position ϑ before one of the wheels starts to slide on the plastic tube; (c)(d) results of the normal force F test.

5

Conclusions

This work focuses on a first functional and mechatronic design and evaluation of a climbing robot for tree pruning operations. In the first section the concept of the adopted solution for the climbing platform, as well as the study of its stability is presented. In the second section the design and the realization of a preliminary version of the robot is shown in details. Finally, in Sect. 4 the experiments that evaluate the effectiveness of the proposed solutions are reported.

Development of a Climbing-Robot for Spruce Pruning

107

Experimental validations show the suitability of the chosen solution in terms of climbing capability, simplicity and lightness. Although a three-wheeled solution is able to guarantee the stability of the platform, wheel side slippage could occur in presence of smaller trunk diameters and thus affect the stability. To avoid and overcome this possible problem, taking also into account the additional tangential load that a cutting element may create, three additional passive wheels could be possibly added as stabilizing elements. A further aspect that would be worth of investigation is the recovery of the system in case of failure. In particular, considering that each wheel gearbox is a reversible transmission (i.e. transmission efficiency higher than 0.5), a passive recovery device by means of an emergency rope could be implemented. Acknowledgement. This work has been supported by the CRC project TN200L “SPRUCE-ROBOT: Smart PRUning and Climbing treEs ROBOT” funded by the Free University of Bolzano and by the project “FiRST Lab: Field Robotics South-Tyrol Lab” (Project number: FESR1084) funded under the European Regional Development Fund (ERDF) 2014–2020 - Province of Bozen (I)-framework.

References 1. Alasky Division of Forestry: Firewise Pruning. Alasky Division of Forestry Publication, December 2007. http://www.forestry.alaska.gov 2. M¨ akinen, H., Verkasalo, E., Tuimala, A.: Effects of pruning in Norway spruce on tree growth and grading of sawn boards in Finland. For. Int. J. Forest. Res. 87(3), 417–424 (2014) 3. Shigo, A.L.: Successions of organisms in discoloration and decay of wood. Int. Rev. For. Res. 2, 237–299 (1967). Elsevier 4. Bob’s tree service. http://www.bobstreeservice.org/Pruning-Mature-Trees.htm. Accessed 12 Feb 2020 5. Advaligno GmbH. https://www.advaligno.com/en/. Accessed 12 Feb 2020 6. Clouston, J.L.: Patent. https://www.google.com/patents/US5983966. Accessed 12 Feb 2020 7. Mustapa M.A., Othman W.A.F.W., Abu Bakar, E., Othman A.R.: Development of pole-like tree spiral climbing robot. In: Intelligent Manufacturing & Mechatronics, pp. 285–293 (2018) 8. Baghani, A., Ahmadabadi, M.N., Harati, A.: Kinematics modeling of a wheel-based pole climbing robot (UT-PCR). In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 2099–2104 (2005) 9. Fu, G., Liu, X., Chen, Y., et al.: Fast-growing forest pruning robot structure design and climbing control. Adv. Manuf. 3, 166–172 (2015) 10. Wibowo, T.S., Sulistijono, I.A., Risnumawan, A.: End-to-end coconut harvesting robot. In: 2016 International Electronics Symposium (IES), pp. 444–449 (2016) 11. Yisheng, G., Li, J., Xianmin, Z.: Mechanical design and basic analysis of a modular robot with special climbing and manipulation functions. In: Proceedings of the 2008 ROBIO Conference, pp. 502–507 (2008) 12. Lam, T.L., Xu, Y.: A flexible tree climbing robot: treebot - design and implementation. In: Proceedings of the IEEE ICRA, pp. 5849–5854 (2011)

108

G. Carabin et al.

13. Li, Y., Chen, M., Chen, Y., Lam, J.: Design of a one-motor tree-climbing robot. In: Proceedings of the 2015 IEEE ICIA, pp. 26–31 (2015) 14. Gasparetto, A., Vidoni, R., Seidl, T.: Kinematic study of the spider system in a biomimetic perspective. In: Proceedings of the 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), art. no. 4650677, pp. 3077– 3082 (2008) 15. Vidoni, R., Gasparetto, A.: Efficient force distribution and leg posture for a bioinspired spider robot. Robot. Auton. Syst. 59(2), 142–150 (2011)

A Suspended Cable-Driven Parallel Robot for Human-Cooperative Object Transportation Yusuke Sugahara1(B) , Guangcan Chen1 , Nanato Atsumi1 , Daisuke Matsuura1 , Yukio Takeda1 , Ryo Mizutani2 , and Ryuta Katamura2 1

Tokyo Institute of Technology, Tokyo, Japan [email protected] 2 Kajima Corporation, Tokyo, Japan

Abstract. This paper describes the development of the suspended cable-driven parallel robot for object transportation having power assist function. The robot consists of the winches to drive four cables which suspend the object and control its three degrees of freedom motion. The impedance controller based on the external force measured by the tension sensors enables the operator to transport and position the object intuitively by directly applying the force to the object. Additionally, the workspace departure avoidance control by installing the virtual stiffness into the controller is also introduced.

Keywords: Cable-driven parallel robot control

1

· Power assist · Impedance

Introduction

In recent years the reduction of the labor force has caused many problems in maintaining social systems. The necessity of productivity improvement is advocated in all industrial fields, and expectations for utilization of robot technology are increasing. Above all, in the construction industry, where labor shortages are serious, the introduction of robot technology is accelerating [1]. Especially, there is a strong demand for labor saving in heavy load transportation and positioning on site. Overhead traveling cranes are commonly used to transport heavy objects, and researches on vibration suppression control have been widely conducted for a long time [2]. However, the stiffness in the horizontal plane is zero in principle, so its automatic positioning is difficult in practice. For the transportation of heavy objects with fine positioning, practical use of systems using robot technology has recently started. ATOUN Model K [3], developed by ATOUN Inc. in collaboration with Shimizu Corp. and SC Machinery Corp., is a robot for assistance in bar arrangement work. This is a serial link type power assist arm that can be mounted on a pillar at a construction site. The operator directly applies force to the bar raised by the robot’s hands, enabling intuitive c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 109–117, 2021. https://doi.org/10.1007/978-3-030-58380-4_14

110

Y. Sugahara et al.

and smooth handling of the target object. This can be considered as an extension of the existing power assist arm called “Balancer” [4] that has been used for more general purposes. Both of these structures are serial link type arm and requires an extremely long link length for the transportation and positioning over a wide area like cranes.

Fig. 1. Conceptual image.

The cable-driven parallel robot (CDPR) is advantageous for such a wide range of transport and positioning [5]. For example, Pott et al., has been investigating a construction system for a solar power plant using CDPR [6]. Above all, the suspended CDPR is considered to have a high affinity for construction sites because its structure is similar to a crane. The application fields of the pioneering suspended CDPR, NIST ROBOCRANE, were the sites of shipbuilding and construction [7]. The superiority over traditional cranes is also claimed in the study of CDPR by Yamamoto et al. in which three mobile trolleys suspend the end effector with parallel cables [8]. The authors’ long-term objective in this study is to develop a suspended CDPR for the transportation and positioning of heavy objects on construction sites, as shown in Fig. 1. This is a suspended CDPR which consists of four winches attached on the upper portion of steel columns in a building under construction like the ATOUN Model K, and four wires reeled out from it. The four wires are connected at one point to form an end effector, and the object is hung at this point. Therefore, the posture of the object is not constrained. The movable range is a cuboid with a rectangle connecting the four wire feeding points as its upper surface, and a wider movable range can be easily obtained than the serial link type. The stiffness in the horizontal plane is higher than the overhead traveling crane, which is advantageous for fine positioning. In addition, this system has a power assist function, and the operator can intuitively transport and position the object by directly applying force to itself. A pioneering study of a similar concept is the Power-Assist Lifting Device [9] by Hayatsu et al. Although the number of wires and the degrees of freedom are

A Suspended CDPR for Human-Cooperative Object Transportation

111

different, this was a suspended CDPR composed of five chain blocks and realized the positioning of heavy objects by power assist using damping control. In this study, the position based impedance control which is more general and its parameter tuning is intuitive is utilized. Furthermore, by introducing a virtual potential field, a control to prevent departure from the workspace is realized.

2

Prototype

The mechanical structure and specifications of the suspended CDPR prototype named TKSC78 (Tokyo Tech Kajima Suspended Cable-Driven Parallel Robot – No. 78) developed in this study are shown in Fig. 2 and Table 1, respectively. As shown in (a) in the figure, this robot is a redundant three degrees-offreedom suspended CDPR in which four cables suspended from four frame anchors on the upper corners of an outer cuboid frame simulating the steel columns in a construction site are connected at one point which is used as the end effector. The winches for winding the cables are fixed at the lower portion Swiveling pulley with built-in tension sensor

Winch

End effector

Chemical fiber wire

Payload

(a) Overall structure Cable

Cable

Spiral pulley

Servo motor

(b) Winch

Tension sensor

(c) Swiveling pulley

Fig. 2. Structure of TKSC78 prototype.

Payload

(d) End effector

112

Y. Sugahara et al.

of the frame, and the winches control the lengths of the cables by driving the spiral pulleys with servo motors as shown in (b) in the figure. The swiveling pulleys shown in (c) in the figure are fixed on the upper portion of the frame as frame anchors, have tension sensors which measure the tension of the cables, and have a function of passively rotating around the Yaw axis according to the cable tension. There is a sufficient distance between the winch and the swiveling pulley so that the fleet angle is less than 1.5◦ , which eliminates the need for a special mechanism to prevent the irregular winding. The end effector is equipped with a hook for hanging the object as shown in (d) in the figure. Table 1. Specifications. Model No.

TKSC78

Length × Width × Heigh 3200 × 6300 × 2000 mm Degrees of freedom Maximum payload Workspace Maximum velocity

3 Translational DOFs 40 kg 3000 × 6000 × 1000 mm 0.5 m/s

The control system is configured using INTime and EtherCAT. The servo driver operates in the position control mode, and position feedback control is performed inside the servo driver. The control program, which is a user application, acquires the angles of all the actuators and the values of the tension sensors in a control cycle of 5 ms, and commands the target angle of each actuator based on the computation by the control algorithm.

3

Controller

In this study, power assist control using position-based impedance control, also known as admittance control, is performed to realize the intuitive transportation and positioning by the operator directly applying force to the object. Furthermore, by extending this method and introducing a virtual potential field, a function to prevent departure from the workspace is realized. The block diagram of the control system is shown in Fig. 3. As an example of using impedance control in CDPR, Osumi et al. proposed a crane with seven actuators which include a suspended parallel cable structure [10], and realized a peg-in-hole task by introducing compliance control [11]. There are also other studies that introduce admittance control to CDPR [12,13]. 3.1

Power Assist Using Impedance Control

In this prototype, as described above, each servo driver operates in a local position control loop, so it has been necessary to set up a position control-based control system. Various methods of force control of robot manipulator have been

A Suspended CDPR for Human-Cooperative Object Transportation

113

Fig. 3. Block diagram of the controller.

proposed, but for this reason, authors decided to use position control based impedance control. Assuming that only the actuation force from cables, the gravitational force and the operation force from operator act on the load, the actuation force from cables fwire ∈ R3 are as follows: fwire = AT τ

(1)

where τ ∈ R4 is wire tension vector, AT is the transpose of the Jacobian matrix referred to as structure matrix [14,15]. When only static transport is considered, the inertial force on the object is negligibly small compared to gravity, so the actuation force fwire , gravitational force G ∈ R3 , and the operating force fext ∈ R3 applied by the operator can be considered to be balanced, and this relationship can be used to estimate the operating force: (2) fwire + fext + G = 0. However, since inertial force is actually applied, a dead zone according to the weight of the object is provided to prevent vibration as follows:  0 (|fext | < λ|G|) fˆext = (3) fext (|fext | ≥ λ|G|) where fˆext ∈ R3 is the estimated value of the operating force with the dead zone applied, λ is the dead zone coefficient. Let the object have a desired impedance characteristic represented by the following equation: (4) M P¨ + C P˙ + KP = fˆext where P ∈ R3 is the position vector of the end effector, M is a virtual inartial matrix, C is a virtual viscous matrix, K is a virtual stiffness matrix, each of which is a diagonal matrix. By discretizing Eq. (4), the acceleration of the end effector at the time of n sampling is as follows: P¨n = M −1 (fˆext − C P˙n−1 − KPn−1 )

(5)

114

Y. Sugahara et al.

where P˙n−1 and Pn−1 are velocity vector and position vector of the end effector at the time of n − 1 sampling. By integrating these, the position vector of the end effector at the time of n sampling is obtained as follows: P˙n = P˙n−1 + ΔtP¨n Pn = Pn−1 + ΔtP˙n

(6)

where Δt is the control cycle. Based on this, the length of each cable is computed by inverse kinematics, and is commanded to each servo driver. This method has the advantage that parameter tuning is intuitive and suitable for site work, and has more expandability than stiffness control and damping control. For example, to realize a power assist to transport an object based on the force applied by the operator, it is enough to introduce only the virtual inertial matrix and the virtual viscous matrix and set the virtual stiffness matrix to 0. But the use of the repulsive force is also effective in some cases as described later. 3.2

Workspace Departure Avoidance Control Using Potential Field

This study proposes a system to transport and position heavy objects according to the operator’s force, rather than transporting them automatically. This method is considered to be effective in a construction site where there are many obstacles and the situation changes every moment. However, since it is not easy for operators to handle huge objects with careful attention to collisions with obstacles and the movable range of the mechanism, the intuitive operation assist control is effective. In this study, for obstacle avoidance and departure prevention from the workspace, avoidance control using the potential method is introduced by using the virtual stiffness of the impedance control parameters. Taking the Y axis direction as an example, let us consider a control which generates a virtual repulsive force when the end effector departs from available space set in consideration of the workspace of the mechanism. This is possible using the virtual stiffness ky whose value depends on the end effector position y, as follows:  0 (LK < y < (Ymax − LK )) (7) ky = K (y < LK , (Ymax − LK ) < y) where LK is the equilibrium length of the virtual spring, Ymax is the length of the edge of the workspace. By introducing this virtual stiffness into the impedance control law described above, as the end effector approaches the boundary of the workspace (Y = 0 and y = Ymax in this example), the virtual potential of the output node increases, so the end effector behaves repulsively from the boundary of the workspace.

A Suspended CDPR for Human-Cooperative Object Transportation

4

115

Experiments

To evaluate the hardware and controller proposed in this study, the object transportation experiment and avoidance control experiment have been done. First, as an object transportation using impedance control, a series of transport operation, in which the end effector is attached to a 5.5 kg object placed on the floor, transported to the target position, removed from the object, and moved to another object position again, was performed. The parameters in this experiment are Cx = Cy = 100, Cz = 40, Mx = My = 5, Mz = 40, λ = 0.03, which have been determined heuristically. Figure 4 shows a series of transportation operations. The operator carried the end effector while holding it directly by hand, and attached the object to the end effector. Next, impedance control was disabled once, the end effector was moved vertically upward to lift the object, and the weight of the object was measured from the value of the tension sensor. After that, impedance control was re-enabled, and the operator carried the object to the target position by applying force directly to the end effector with one hand, and positioned it. Next, impedance control was disabled again, and the object was moved vertically downward to ground the object. Thereafter, the same procedure was used to return the end effector to its original position. It was confirmed that it was able to smoothly transport and position the object of a weight that would not normally be transported by one operator’s one hand by using the developed system.

t=0s

t = 20 s

t = 30 s

t = 45 s

t = 52 s

t = 64 s

t = 73 s

t = 92 s

t = 118 s

t = 129 s

t = 140 s

t = 151 s

Fig. 4. Object transportation and positioning.

Furthermore, Fig. 5 shows the operator intentionally pushing the object toward the workspace boundary. It can be seen that the object is pushed back into the workspace when the operator releases his/her hand. This is useful as a safety function to prevent the operator from inadvertently pushing an object out of the workspace. Using this method, it is also possible to avoid obstacles during operation by installing a potential field appropriately near obstacles.

116

Y. Sugahara et al.

t=0s

t=2s

t=4s

t=6s

t=8s

t = 10 s

Fig. 5. Avoidance motion.

5

Conclusions and Future Work

In this paper, the development of a suspended CDPR for object transportation and positioning with a long-term objective of application on construction sites was presented. The design is based on a three degrees-of-freedom suspended CDPR that controls the position of an object by hanging with four cables. By the impedance control using cable tension sensors, intuitive transportation and positioning by the operator directly applying force to the object were achieved. Furthermore, by extending this control, workspace departure avoidance control was also introduced. Authors’ future work before on-site application includes the performance evaluation using external measurement devices, the vibration suppression control and automatic generation of potential field with obstacle detection. Acknowledgment. This work was partially supported by JKA and its promotional funds from the AUTORACE and KEIRIN RACE.

References 1. Delgado, J.M.D., Oyedele, L., Ajayi, A., Akanbi, L., Akinade, O., Bilal, M., Owolabi, H.: Robotics and automated systems in construction: understanding industry-specific challenges for adoption. J. Build. Eng. 26 (2019) 2. Lee, H.H.: Modeling and control of a three-dimensional overhead crane. J. Dyn. Syst. Measur. Control (1998) 3. ATOUN Inc.: ATOUN MODEL K. http://atoun.co.jp/products/atoun-model-k 4. TOYO KOKEN K.K.: Balancer. https://www.toyokoken.co.jp/english/products/ balancer/ 5. Gosselin, C.: Cable-driven parallel mechanisms: state of the art and perspectives. Mech. Eng. Rev. 1(1) (2014) 6. Pott, A., Meyer, C., Verl, A.: Large-scale assembly of solar power plants with parallel cable robots. ISR 2010 and ROBOTIK 2010 (2010) 7. Albus, J., Bostelman, R., Dagalakis, N.: The NIST ROBOCRANE. J. Res. Nat. Inst. Stand. Technol. 97(3) (1992) 8. Yamamoto, M., Yanai, N., Mohri, A.: Trajectory control of incompletely restrained parallel-wire-suspended mechanism based on inverse dynamics. IEEE Trans. Robot. 20(5) (2004)

A Suspended CDPR for Human-Cooperative Object Transportation

117

9. Hayatsu, M., Yamada, M., Tagawa, Y., Yamaguchi, D.: Development of a powerassist lifing device using a multi-DOF suspension mechanism (1st report, study of a power-assist lifting device and its control technique). Trans. Jpn Soc. Mech. Eng. Ser. C 68(671) (2002). (in Japanese) 10. Osumi, H., Arai, T.: Heavy object handling system by cooperation of a robot and a crane with multiple wires. In: Proceedings of the 13th ISARC (1996) 11. Osumi, H., Hashimoto, G., Sugihara, M.: Compliance control of a parallel wire crane. J. Jpn Soc. Precis. Eng. 66(5) (2000). (in Japanese) 12. Rezazadeh, S., Behzadipour, S.: Impedance control of cable-driven mechanisms. In: Proceedings of the ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2008) 13. Ho, W., Kraus, W., Mangold A., Pott, A.: Haptic interaction with a cable-driven parallel robot using admittance control. In: Cable-Driven Parallel Robots (2015) 14. Roberts, R.G., Graham, T., Trumpower, J.M.: On the kinematics and statics of cable-suspended robots. In: IEEE International Conference on Systems, Man, and Cybernetics (1997) 15. Pott, A.: Cable-Driven Parallel Robots - Theory and Application. Springer, Cham (2018)

Automatic Planning of Psychologically Less-Stressful Trajectories in Collaborative Workstations: An Integrated Toolbox for Unskilled Users Rafael A. Rojas(B) , Manuel A. Ruiz Garcia, Luca Gualtieri, Erich Wehrle, Erwin Rauch, and Renato Vidoni Faculty of Science and Technology, Free University of Bozen-Bolzano, Bolzano, Italy {rafael.rojas,ManuelAlejandro.RuizGarcia,luca.gualtieri,erich.wehrle, erwin.rauch,renato.vidoni}@unibz.it

Abstract. Collaborative robotics relies on a safe physical interaction between robot and operator. However, even if compliance with safety standards is assured, other kind of dangers like psychological stress can be neglected or improperly considered. It has been proven that working and sharing workspace with robots can be psychologically stressing for the operator. Further, small and medium-sized enterprises inherently lack highly skilled workers in. These two factors are identified as risks to the acceptance and usage of collaborative robots. In order to overcome these issues, we developed, implemented and validated a new toolbox and graphical user interface for collaborative robots to allow unskilled workers to easily and automatically plan psychologically less-stressful trajectories. This method assures smooth movements and maximum allowable speeds. This work introduces the toolbox with collaborative robots of Universal Robots, though is general and applicable to other manufacturers. Keywords: Collaborative robotics · Trajectory planning Psychological stress · Cognitive ergonomics

1

· Safety ·

Introduction

Industry 4.0 technologies are driving the evolution of both large as well as small and medium-sized enterprises (SME) [10]. Collaborative robots (cobots) represent a cornerstone of these technologies, which are gaining attention in research and on the market. Cobots are conceived and designed for safe physical humanrobot interaction (pHRI) according to the technical specification TS-15066 [3]. These robotic systems therefore limit displacement, force and power. Further, they allow a simple, intuitive and fast programming phase due to the fact that also unskilled workers, e.g. in particular in SMEs, are and will be employed in collaborative activities. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 118–126, 2021. https://doi.org/10.1007/978-3-030-58380-4_15

Automatic Planning of Psychologically Less-Stressful Trajectories

119

The most adopted programming and motion planning technique is based on a hand-guiding method: the robot is moved manually into a sequence of desired positions, also called via-points. This delegates the traditional difficulties of the path planning problem to the controller. Once the operator has taught the set of desired via-points, the robot controller plans the trajectory using algorithms that rely on primitives often based on trapezoidal speed profiles and safety limits. Although this method results simple and transparent to the user, the final result does not take into account possible psychological stress that could be induced to employees when working with robots. Indeed, it has been proven that humans perceive subjective levels of a hazard, chance of error and mental workload with the presence of moving robots even if they are at safe distances [6]. The mental strain produced in collaborative robotics stations with a medium-sized robot has been studied in [1] using point-to-point trapezoidal velocity profile trajectories. These authors concluded that robots induce high mental strain even at large distances moving at velocities compliant to the TS-15066 [3]. A further work measured the perceived safety and comfort of the operator in a collaborative station [5]. In [4], the authors use point-to-point minimum-jerk trajectories to generate psychologically “acceptable” motions without inducing a disturbing or uncomfortable feeling to the operators. From these studies, it can be derived that movements that can reduce the psychological stress may be achieved by predictable smooth motions based on minimum-jerk trajectories. Moreover, a second major result can be obtained via smooth trajectories: they are mechanically gentle and allow to reduce induced vibrations and wear in the robot mechanics [2]. Based on this evidence, we can assume that hand-guiding methodologies for the generation of minimum-jerk motions that would allow a reduction of psychological stress would be beneficial for the introduction of collaborative robots in SMEs. However, the complexity of the minimum-jerk motion algorithms and their implementation forbids unskilled users to implement less stressful motions in co-bots thus possibly preventing SMEs to exploit their benefits. In this work, we try to fix this gap by creating and integrating in a cobot controller and teaching pendant a new user-friendly toolbox and graphical user interface (GUI) for unskilled users in order to allow them to automatically plan minimum-jerk trajectories for the reduction of psychological stress while using common hand-guided methods. After the introduction of the theoretical background of the implemented minimum-jerk trajectories based on the method developed by the authors in [9], we present the design and implementation of the PSafe toolbox for UR robots [7] as well as the results achieved on a collaborative assembly station.

2

Motion Planning for Collaborative Robots

Smooth trajectories are obtained when the jerk (absolute value of the derivative of the acceleration) at the joints remains bounded. However, the standard motion planning methods embedded in a collaborative robot are usually based on trapezoidal speed profile primitives, and therefore exhibits discontinuities in the jerk.

120

R. A. Rojas et al.

Starting from the results presented in [9], the authors have further developed a new trajectory-planning methodology and representation based on the calculus of variations to identify a family of feasible minimum-jerk trajectories. Among the suitable family of trajectories, the fastest trajectory can be chosen. 2.1

Minimum-Jerk Trajectory Based on a Variational Approach

We describe the trajectory of the robot as a configuration space or a given task space in Z = [0, T ], we search for a trajectory q : Z functional is minimized:  T ... 2 I(q) = q

curve in the space in either the Rn . Given a fixed time interval −→ Rn such that the following dt,

(1)

0

where T is an arbitrary execution time. This equation is subject to the condition of passing through a series of N + 1 via-points {q0 , q1 , · · · , qN } at a series of unknown time instants {t0 , t1 , · · · , tN }, where q0 and qN are the first and last points of the trajectory attained at the initial and final times, i.e. t0 = 0 and tN = T . These constraints as well as those on velocity and acceleration at ˙ 0 ) = q˙ 0 , q(t ˙ N ) = q˙ N , t0 = 0 and tN = T can be then written as: q(ti ) = qi , q(t ¨ 0 and q ¨ (tN ) = q ¨ N . To identify the trajectory q that is a local extremal ¨ (t0 ) = q q for the functional in Eq. (1) and following the calculus of variations formalism, we define the family of variations of q given by p = q(t) + εη(t), where η ∈ C ∞ (Z, Rn ) is an arbitrary map called variation and ε is an arbitrarily small scalar. We enforce the following necessary optimality condition:  ∂  I(p) = 0 ∂ε

(2)

As demonstrated in [9], the optimal solution does not have to be six times continuous differentiable on the whole interval Z. By introducing the vector τ ∈ RN which components are τi = ti+1 − ti , the solution can be represented as: qi (t) = (yij ) B(sj (t)) if t ∈ Zj ,

(3)

where B represents the vector defined by stacking a suitable basis of the 6 vector space of fifth-order polynomials {Bi }i=1 and yij ∈ R6 represents the six coefficients of the polynomials of the component i at the interval Zj . It is therefore possible to introduce the vector y = [(y11 ) , · · · , (ynN ) ] of dimension np = 6nN , representing all the coefficients  of the interpolating polynomials.

Substituting (3) in (1) results in I = 32 y Q(τ )y , where Q(τ ) is a block diagonal matrix. Furthermore, the via-points and continuity constraints may be written as: A(τ )y = b, (4)

Automatic Planning of Psychologically Less-Stressful Trajectories

121

where A(τ ) is a 6nN × 6nN invertible matrix and b is a vector containing the via-points and the boundary conditions. Thanks to (4) the problem becomes:     1 τ = T,  − −1 s.t. min I(τ ) = min b A (τ )Q(τ )A (τ )b τi > 0, for i = 1, . . . , N τ τ (5) where 1 ∈ RN is the all-ones column vector. 2.2

The Invariance Property of the Extremals

The execution time in (1), if not known a priori, may represent a major problem for the trajectory planning. Indeed, if such a time is not given, the problem stated above could seem incomplete. However, this is not the fact, as the extremals of (1) have the property of being invariant with respect to linear time dilations. Let us consider the linear dilation which dilates the execution of a trajectory from the interval Z = [0, T ] to the interval Z  = [0, T ]. We can refer to such as a map as σ : C(Z, Rn ) −→ C(Z  , Rn ), which is defined by σ(q) = q s.t. q (s) = q(σ0 t) , σ0 ∈ R+ , s ∈ Z  , where T = σ0 T . Such a dilation transforms the integral (1) into  T /σ0 ... 2   5  q  dt , I(q ) = σ0 (6) 0 



where q = σ(q) and t = t/σ. An analysis of such an integral reveals that the extremal points of (6) must respect the same stationarity conditions of the original problem, i.e. if q is an extremal, σ(q) respects all the optimality conditions. It turns out that computing an extremal in an arbitrary interval, e.g. [0, 1], allows to find a one-parameter family of extremals for an execution time T.

3 3.1

Development of a Toolbox for Non-skilled Users UR Control Architecture

We consider robots from Universal Robots (UR [7]) which produces 6-axis anthropomorphic cobots with a non-spherical wrist. This robot has an almost spherical workspace with radius from 500 mm to 1300 mm and is controlled by a Mini-ITX motherboard and a safety PLC. The PLC handles the implementation of the pHRI features of the robot and it represents the gateway to the internal bus of the robot arm. The Mini-ITX installed on these robots runs a Linux operating system and is connected to the teaching pendant where the GUI Polyscope is provided. In order to develop custom applications for UR, two main possibilities are available: (a) a script language called Polyscript and (b) a visual version of this language based on a three-node paradigm that we will call URP format. Polyscope gives the operator access to the robot functionalities and provides an integrated development environment (IDE) for programming in

122

R. A. Rojas et al.

URP format. Each URP program is composed by installation nodes, program nodes and daemon executables: (i) installation nodes provide the means to setup general features of the robot as safety parameters and hardware devices; (ii) program nodes are single actions of the robot; (iii) daemon executables are applications which run in the Linux background and provide functionalities that are not provided by Polyscope. Before executing a URP program, the Polyscope IDE translates it into Polyscript. The Polyscript language is interpreted by the UR-control daemon which interacts with the safety PLC and the robot’s arm, see Fig. 1. In addition, Polyscript allows basic TCP/IP socket communication and supports up to some degree the XML-RPC remote procedure call method to allow an application interact with external programs. This communication feature allows to create an internal bus inside the Mini-ITX to provide interoperability between different applications.

Internal Bus

Mini ITX PC

Touch Pendant

Daemon 1 Polyscope Daemon 2

Safety PLC URControl Daemon n

Robot Arm

Ethernet Based external bus

External system

Fig. 1. Control architecture of the UR family of robots.

To expand the capabilities of the URP language, Polyscope provides modular packages called URcaps. URCaps permit to integrate any customized functionality into the Polyscope IDE and allows the developer to use a high level language as JAVA to define screens and program nodes for the end user. Thus, it is possible to encapsulate new complex behaviours or provide friendly hardware interfaces. A URCap package consists of a Java Archive ( .jar ) file with the .urcap file extension. It contains all the information required to define one or more installation nodes, program nodes and daemon executables. Each URCap is composed by a HTTP piece of code providing the layout and geometry of the visual interface, a Service interface that makes the node available to PolyScope and the contribution containing the main functionalities of the URCap.

Automatic Planning of Psychologically Less-Stressful Trajectories

3.2

123

Psafe-Toolbox and GUI

Fig. 2. Screenshot of the developed Psafe-toolbox and GUI: (left) node of the MJT planners; (right) installation node.

The software package enabling robot to execute motion is described in Fig. 2. It has client–server architecture composed by two separate modules, a user interface and an optimizer. The first module is written in Python 3 and provides a solver for the problem stated in Sect. 2 using the second-order optimizer IPOPT. This package is to be installed on an external computer and it provides an API based on TCP/IP sockets. The second module is a URCap module called MJT planner : it provides a user interface to introduce the via-points taught through the hand-guided programming phase and it is connected with the optimizer API. The program node and the installation window of the MJT planner are presented in Fig. 2. Thanks to this software, the user is able to work and plan the robot motion as before but with the chance to choose between the “normal” operation mode or the psychologically less-stressful one, i.e. minimum-jerk motion. In addition, to improve the safety of the station, the user can also introduce a velocity limit of the end-effector vad . This allows to scale the planned trajectory, if needed, after performing a mechanical risk assessment of the collaborative station under evaluation. Indeed, thanks to the invariance property of the minimum-jerk motions planned with our method, we can compute an optimal motion through the given via-points for an arbitrary execution time T . The dilation factor is then comad = σ0 , where vmax is the maximum linear velocity of the endputed as: vvmax effector within the planned trajectory.

4

Validation and Discussion

The developed system has been implemented and tested on a UR3 co-bot. The chosen test-case is related to a collaborative workstation that has been realized starting from an existing manual assembly workstation of pneumatic cylinders, see [8]. Both the manual and collaborative workstations are depicted in Fig. 3.

124

R. A. Rojas et al.

Fig. 3. Test-case: (left) original manual assembly station; (center) collaborative station based on the UR3 robot; (right) via-points teaching procedure.

To plan the path and the trajectory, the user/worker moves in free-drive the robot and teaches it the sequence of chosen via-points manually. Thanks to the Psafe-GUI, the user selects the kind of motion to be planned and, in particular, he/she has the chance to select and plan a min-jerk trajectory computed with the method presented in this work without differences with respect to the standard modality, i.e. in a transparent way. To show the developed method, a pickand-pass task composed of two different trajectories executed in sequence, the first for picking and the second for passing, is here presented and the results of both the native robot planner and our developed method compared. The first phase is a hand-guiding programming one in which the sequence of viapoints reported in Table 1 has been stored. At this point the user selects which kind of trajectory he would like to perform thanks to the developed toolbox and it is computed and planned accordingly. In this specific example, the total execution time of the minimum jerk motion was 21.5 s and the embedded robot planner was 19.9 s. In terms of complexity for the human, since the planning procedure does not change, there are no differences. Concerning the trajectory smoothness, i.e. “psychologically” safe trajectory, the joint space trajectories planned with both our method and the one embedded in the robot planner are depicted in Fig. 4. Here, the difference of smoothness between both motions can be appreciated. Such an improvement in smoothness and, thus, such a possible decrease of psychological stress, comes at the cost of a slight increase of the execution time and the possible appearance of small overshoots along the motion. Table 1. Via-points (radiants): (left) picking trajectory; (right) passing trajectory. J 1

J 2

J 3

J 4

J 5

J 6

J 1

J 2

J 3

J 4

J 5

J 6

q0 −1.653 −1.920 −1.995 −2.442 −1.643 1.584

0.903 −1.993 −2.393 −1.951 −1.983 1.613

q1 −0.407 −1.644 −2.334 −2.390 −1.561 1.583

0.903 −1.993 −2.393 −1.951 −1.983 1.613

q2 −0.410 −1.929 −2.468 −1.893 −1.694 1.582

0.909 −1.634 −2.286 −2.434 −1.977 1.606

q3 −0.161 −1.855 −2.593 −1.854 −1.863 1.561

0.604 −1.634 −2.286 −2.434 −1.006 1.606

q4

0.145 −1.835 −2.632 −1.844 −2.020 1.558 −0.023 −1.634 −2.286 −2.434 −1.379 1.606

q5

0.532 −1.917 −2.443 −1.933 −2.146 1.563 −0.660 −1.621 −2.285 −2.434 −1.977 1.606

q6

0.706 −1.918 −2.531 −1.894 −2.181 1.600 −0.644 −1.384 −2.035 −2.912 −0.349 1.596

q7

0.903 −1.993 −2.393 −1.951 −1.983 1.613 −1.417 −1.849 −1.502 −2.955 −1.122 1.561

Automatic Planning of Psychologically Less-Stressful Trajectories [rad]

[rad/s]

Joint 1

[rad]

[rad/s]

Joint 2

1 1

0.5

−1.4

[rad]

125 [rad/s]

Joint 3

−1.5 0.5

0

−1.6

0

−2

0

0 −1.8

−1

−0.5 −0.5 −2.5

−1 0

5

10

15

20

Time [s]

−2

−1 0

5

10

Time [s]

15

20

0

5

10

15

20

Time [s]

Fig. 4. Comparison between the minimum jerk (solid lines) and robot’s “native” trajectories (dashed lines) joining the via-points in Table 1, first three joints: (blue) joint positions; (red) joint velocities.

The minimization of the jerk implies that a trajectory preserves a given value of its acceleration for a longer time with respect to trapezoidal velocity profile. Then, to make the motion mechanically safe, larger trajectory scaling factors are to be applied with respect to the trapezoidal velocity profile.

5

Conclusions

In this paper we describe the implementation of a robot-programming toolbox in which easily generate trajectories that induce less stress in operators. The human-robot interaction has thus been improved with respect to trajectories created through the native motion planner, i.e. trapezoidal velocity profile trajectories. Thanks to the invariance property of the minimum-jerk motion with respect to linear dilations of the execution time, we are able to ensure physically safe motion via time-scaling techniques. As a proof of concept, we present an experimental setup where the PSafe toolbox has been implemented and provide a comparison between the different methods. Future works will address extensive tests on the comfort perceived by the operator.

References 1. Arai, T., Kato, R., Fujita, M.: Assessment of operator stress induced by robot collaboration in assembly. CIRP Ann. 59(1), 5–8 (2010) 2. Gasparetto, A., Zanotto, V.: A new method for smooth trajectory planning of robot manipulators. Mech. Mach. Theory 42(4), 455–471 (2007) 3. Robots and robotic devices – collaborative robots. Technical specification, International Organization for Standardization, Geneva, CH (2016) 4. Kokabe, M., Shibata, S., Yamamoto, T.: Modeling of handling motion reflecting emotional state and its application to robots. In: Proceedings of the SICE Annual Conference, pp. 495–501. IEEE (2008) 5. Lasota, P.A., Shah, J.A.: Analyzing the effects of human-aware motion planning on close-proximity human-robot collaboration. Hum. Factors 57(1), 21–33 (2015)

126

R. A. Rojas et al.

6. Or, C.K., Duffy, V.G., Cheung, C.C.: Perception of safe robot idle time in virtual reality and real industrial environments. Int. J. Ind. Ergon. 39(5), 807–812 (2009) 7. Robots, U.: User manual ur3/cb3 original instructions. Technical reports, Universal Robots 8. Rojas, R.A., Rauch, E., Dallasega, P., Matt, D.T.: Safe human-machine centered design of an assembly station in a learning factory environment. In: Proceedings of the 8th International Conference on Industrial Engineering and Operations Management (2018) 9. Rojas, R.A., Garcia, M.A.R., Wehrle, E., Vidoni, R.: A variational approach to minimum-jerk trajectories for psychological safety in collaborative assembly stations. IEEE Robot. Autom. Lett. 4(2), 823–829 (2019) 10. Rojas, R.A., Rauch, E., Vidoni, R., Matt, D.T.: Enabling connectivity of cyberphysical production systems: a conceptual framework. Procedia Manuf. 11, 822– 829 (2017)

A New Method of Climbing Downstairs by Changing Layers of Gears of Planetary Wheels for Wheelchair Tian-ci Jiang and Eiichiro Tanaka(B) Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsu-ku, Kita-Kyushu, Fukuoka 808-0135, Japan [email protected]

Abstract. As the number of elderly people increases and their leg muscles deteriorate, wheelchairs have become an important means of their transportation. In previous research, we proposed a design that combines a double planetary gears’ wheels with the crawler mechanism to enable wheelchairs to climb upstairs and downstairs. However, while climbing downstairs the situation is different from climbing upstairs. The double planetary gears’ wheels cannot be provided with sufficient friction. Thus, the carrier of the planetary gears cannot revolute as if climbing upstairs and the moving on the stairs will be not stable. To solve the problem, we have designed single planetary gears’ wheels to drive the wheelchair. Both theoretical calculations and proportional models have proved that it will generate fewer accelerations than the previous way, which will ensure a safer and smoother process of climbing downstairs. Keywords: Wheelchair · Stair-moving · Mechanism · Assistance

1 Introduction A large number of disabled people with deteriorating leg muscles emphasize the need for means of transportation. One way to eliminate mobility problems is to use a wheelchair. It is indicated that Japan’s wheelchair sales have reached 4 million [1]. Although wheelchairs can help people with a muscular weakness to move, their daily lives remain difficult. Currently, most wheelchairs cannot cross obstacles or stairs [2]. In order to provide a more convenient means of transportation and reduce the burden on support staff, wheelchairs that can move on stairs with good performance are of great significance and value. A new mechanism with a simple structure is proposed [3], which has the advantages of double planetary gears’ wheels [4] and a crawler mechanism as shown in Fig. 1. The double planetary gears’ wheels mechanism can climb stairs efficiently in the front, and the rotatable crawler can adapt to stairs of different sizes, which will reduce the vibration of the device when moving on the stairs, thereby ensuring a safer moving process. However, in the previous researches, during the process of climbing upstairs, the stairs will provide enough friction to ensure that the device climbs upstairs smoothly © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 127–135, 2021. https://doi.org/10.1007/978-3-030-58380-4_16

128

T. Jiang and E. Tanaka

Fig. 1. Proposed mechanism can climb stairs of different sizes efficiently and stably.

Fig. 2. Climbing upstairs and downstairs with double planetary gears’ wheels.

because the planetary wheels will touch the vertical sections of stairs tightly. But while climbing downstairs, the wheels will not be able to touch the vertical sections of the stairs, which will greatly affect the stability of climbing downstairs. The previous approach was to slow down the speed so that small wheels can cling to the stairs. However, it remains the danger to fall down. A safer and more reliable way to climb downstairs should be proposed.

2 Newly Proposed Mechanism 2.1 Previous Mechanism for Climbing Upstairs and Downstairs We proposed two mechanisms in [3] to compare them with the mechanism consisting of planetary wheels both for front and rear wheels as shown in Fig. 3. Mechanism I is consisting of planetary wheels and fixed crawler, mechanism II is consisting of planetary wheels and rotatable crawler as we introduced just now, and mechanism III is consisting of planetary wheels both for front and rear wheels. We used double planetary gears’

A New Method of Climbing Downstairs

129

wheels to climb the stairs. Firstly, we showed the method to climb upstairs as shown in Fig. 2(a) and Fig. 2(b). While climbing upstairs, the front small wheel touches the stairs and then is fixed. The whole carrier begins to revolute which is very stable. While climbing downstairs, it will not be provided with sufficient friction to cause the entire carrier to roll over, so the entire device will fall directly downstairs, which is very unstable as shown in Fig. 2(c). In previous researches, we compared the performances of three mechanisms. We compared their angle change θ while climbing upstairs and downstairs on stairs of 30°. The green zone means the entire mechanism is moving on the stairs. We conducted both the theoretical calculations and scaled models’ experiments. From the results, we could see that the performance of mechanism II is the best with the smallest vibration while climbing upstairs and downstairs both for theoretical calculations and experiments. For mechanism I, because of the truth that the crawler is fixed while it is not able to adjust itself to the stairs, so the trajectories are not so stable. For mechanism III, because of the sizes of stairs, the front planetary wheels and the rear planetary wheels cannot be always in the same phase of revoluting. Thus, the extra vibration will be generated. Comprehensively we chose the mechanism II.

Fig. 3. Comparisons of three mechanisms of theoretical calculations and experiments.

However, we hoped that the process of climbing downstairs could be in the same way as climbing upstairs, which means that the small wheel can touch the vertical and horizontal sections of stairs tightly and the carrier can revolute entirely. Thus, we considered a new approach to climbing downstairs. 2.2 New Mechanism for Climbing Downstairs Here we showed the new method to climb downstairs as shown in Fig. 4. In the new structure, we changed the previous double-layer gears’ structure to a single-layer gears’

130

T. Jiang and E. Tanaka

structure. This structure has many advantages while climbing downstairs. When climbing downstairs, the lowest small wheel will touch tightly to the horizontal and vertical sections of stairs tightly, so the carrier of the planetary wheels can be turned over entirely as shown in Fig. 4b. When the action on the next stair begins, even if the small wheel that touches the next stair is not close to the vertical sections of the stairs after the carrier rolls over, it can move back and touch the vertical sections of the stairs due to the driving of the single planetary gears. Therefore, this mechanism achieves the position compensation to a certain extent.

a. Single planetary gears’ wheels

b. Process of climbing downstairs

Fig. 4. Climbing downstairs with single planetary gears’ wheel.

The transmission diagram of gears of the new method is shown in Fig. 5. To convert the processes between climbing upstairs and downstairs, the blue parts are designed to be movable. While climbing upstairs, the blue parts will drive the double planetary gears’ wheels to realize the process of climbing upstairs. While climbing downstairs, the blue parts will be moved to drive the single planetary gears’ wheels to realize the process of climbing downstairs.

a. Transmission method for climbing upstairs b. Transmission method for climbing downstairs

Fig. 5. Transmission diagram of gears for climbing upstairs and downstairs.

3 Theoretical Calculation and Experiments of Scaled Models First, we calculated the theoretical trajectories of the center of the planetary wheels’ carrier and the center of the last wheel of the crawler mechanism. As for the crawler mechanism, for its automatic adjusting to the stairs, we supposed that it’s a straight line.

A New Method of Climbing Downstairs

131

For the trajectory of the planetary wheels’ carrier, the moving process is shown as Fig. 6. It consists of four steps. H, W, r, R, L mean height and width of stairs, radius of the small wheel, length of the carrier, distance between two small wheels. θ 1 , θ 2 , L 1 , L 2 mean some important angles and distances while climbing downstairs. (x 1 , y1 ), (x 2 , y2 ) mean the coordinates of the two small wheels touching the stairs tightly and (x 3 , y3 ) means the coordinates of the center of the carrier. By Eq. (1) to Eq. (4), the trajectory of the small wheel from P0 to P1 and P1 to P2 could be calculated.

Fig. 6. Trajectories calculations of the single planetary gears’ wheels for climbing downstairs.

L=

√

3R

 r  L1 = W + r − Lcos arcsin( ) − r L θ1 =

L1 r

L2 = H − r

(1) (2) (3) (4)

Coordinates of the center of the carrier could be calculated by Eq. (5) and Eq. (6).   π y2 − y1 x3 = x1 + R cos − + arctan( ) (5) 6 x2 − x1   y2 − y1 π y3 = y1 + R sin − + arctan( ) (6) 6 x2 − x1 Coordinates from point P2 to P3 could be calculated by Eq. (7).   H 2 θ2 = π − arcsin 3 L

(7)

The results of trajectories are shown as the red dot and the blue dot showed respectively in Fig. 7a. By drawing lines between two lines of trajectories, we could obtain the entire angle of the mechanism and conduct further information. The definitions of angle θ and direction of acceleration a of the mechanism are shown in Fig. 7b. Only the acceleration whose direction is vertical to the stairs was taken into account for it was the biggest. We considered the weighted acceleration which can evaluate the human reaction to different intensities of vibration while accessing transportation in public places [5].

132

T. Jiang and E. Tanaka

a. Theoretical calculations of trajectories with single planetary gears’ wheels b. Definitions of angle and acceleration direction of the mechanism

Fig. 7. Theoretical calculations of the mechanism while climbing downstairs with single planetary gears’ wheels.

We used Eq. (8) and Eq. (9) to calculate the integral acceleration of the orange point in Fig. 7b of the mechanisms given as: 

1 aw = T 

1 aθ = T



T

a (t)dt 2

 21 (8)

0



T

θ (t)dt 2

 21 (9)

0

where aw is the integral acceleration, aθ is the integral angular acceleration, T is the measuring time when the whole equipment is moving on the stairs, and a is the acceleration at each time of the orange point. Since the apparatus moves very slowly on the stairs, we did not consider the acceleration in the direction of movement and we assumed that the apparatus did not turn while moving on stairs, so we only considered the acceleration in the vertical direction. We calculate the acceleration of the orange point using Eq. (10) given as: a = θ¨ L3

(10)

L 3 means the half-length of the frame. According to the acceleration evaluation method [5], we used Eq. (11)

1 2 as = aw2 + 0.22 aθ2 + 0.42 aw2 + 0.42 aw2

(11)

to calculate the weighted acceleration as that evaluates if the transportation is comfortable for the human occupying the seat. Then, to prove the correction of the theory, we made a scaled model and built scaled stairs to conduct the experiments as Fig. 8a shown. The ratio of the model to the assumed prototype is 1: 4. The angle of stairs is 30°. We performed three experiments and captured the process using a camera. Kinovea software enables us to capture two important points of the planetary wheels and crawler mechanism. We connected these points and measured the angle change relative to the ground to the time as shown in Fig. 8b and deduced the acceleration.

A New Method of Climbing Downstairs

133

Fig. 8. Experiment of the scaled model for climbing downstairs with single planetary gears’ wheels.

The angle change to the time of the mechanism is shown in Fig. 9a. We adjusted the time to correspond to the previous method to obtain the acceleration with the same speed of climbing downstairs. The results of the experiments are shown with average values. We could see that the trajectories of the angle with single planetary gears’ wheels are smoother both in theory and experiments. From the results shown in Fig. 9b, we can see that the acceleration of the new method has been reduced both in theoretical calculations and experiments which proved the more stable process for climbing downstairs with the new mechanism. We also considered the situations on different stairs. The sizes of stairs referred to normal sizes of stairs in public places [6]. As the results suggested in Fig. 10 and Table 1, the mechanism climbed downstairs in the new method with a less weighted acceleration which proved the comfort and stability on different sizes of stairs. According to the value 0.315 [m/s2 ] under which the human will not feel uncomfortable while taking transportation, we could adjust the speed of our mechanism to improve the comfort of the user further.

Fig. 9. Comparisons between theoretical calculations and results of experiments.

134

T. Jiang and E. Tanaka

Fig. 10. Comparisons between double and single planetary gears’ wheels on different stairs. Table 1. Values of accelerations between double and single planetary gears’ wheels.

Weighted acceleration as [m/s2 ]

Mechanism

Max

Min

Average

Double planetary gears’ wheels

0.99

0.59

0.76

Single planetary gears’ wheels

0.83

0.42

0.64

4 Conclusion We proposed a new method for the stair-moving able wheelchair consisting of planetary wheels and crawler mechanism to climb downstairs in a more stable way. In the new method, the carrier of the planetary wheels will revolute and realize the process of climbing downstairs more stably with single planetary gears’ wheels. Both the theoretical calculations of trajectories and the results of the experiments of the scaled model showed that in the new method the mechanism could climb downstairs with less weighted accelerations on different stairs, which proved the stability of the new method. As future work, we will make the prototype to evaluate its effectiveness.

References 1. Yamashita, S., Tanaka, S.: Walking support function and safety of silver cars and walking cars. J. Biomech. Soc. 39(3), 124 (2015). (in Japanese) 2. Ikeda, H., Kanda, H., Yamashima, N., Nakano, E.: Step climbing and descending for a manual wheelchair with a network care robot. In: The Second International Conference on Intelligent Systems and Applications, Venice, Italy, p. 95 (2013) 3. Jiang, T., Yin, S., Tanaka, E.: Wheelchair able to assist the elderly to move on stairs and stand up. In: 2019 58th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), Hiroshima, Japan, pp. 1168–1173 (2019) 4. Suzuki Gokin CO., LTD: Traveling vehicles such as wheelchairs, traveling vehicles for the elderly, and transport carts equipped with stair-moving traveling wheels. Japanese patent registration number: 3001961 (1994). (in Japanese)

A New Method of Climbing Downstairs

135

5. Maeda, S.: Trends of vehicle vibration evaluation according to the ISO 2631 standard. J. Acoust. Soc. Jpn 53(1), 33–38 (1997). (in Japanese) 6. Mori, T., Yoshimura, M., Sugano, Y., Muraki, S.: The effects of stair dimensions on movements during climbing upstairs and downstairs. J. Biomech. Soc. 35(3), 201 (1997). (in Japanese)

Design and Analysis of Cable-Driven Parallel Robot CaRISA: A Cable Robot for Inspecting and Scanning Artwork Philipp Tempel1(B) , Matthias Alfeld2 , and Volkert van der Wijk1 1

2

Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands [email protected] Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands

Abstract. Cultural heritage science envisages understanding of methods and techniques used by past painters and sculptors in creating their masterpieces of art. Existing devices for in situ and non-destructive, automated scanning are large and bulky and built around the assumption of a perfectly planar surface. We are developing a lightweight, portable robot for scanning of paintings, marbles, or statues while explicitly allowing for their out-of-plane surface. This paper presents the kinematic design and analysis of the wrench-feasible workspace of a cable-driven parallel robot capable of positioning an imaging device with three translational and two rotational degrees of freedom. At the end stand geometric parameters optimized for the application requirements allowing for pan and tilt of 70◦ each in total, making scanning of the spatial surface of art objects possible.

Keywords: Cable-driven parallel robot Workspace analysis

1

· Geometric design ·

Introduction

Cultural heritage science aims at deeply understanding cultural heritage objects which necessitates knowledge of its material character reflecting creation and history. Insight is commonly acquired by means of imaging techniques such as Xray fluorescence imaging or reflectance imaging spectroscopy for the identification of chemical properties which can reach every layer of paint of an image, even those below the visible surface. Attribution of paintings to famous artists e.g., Vincent van Gogh is one use case [3], preservation for future generations another. Given the heterogeneity of a cultural heritage object, it is often not sufficient to investigate selected spots, but to scan a significant part of the object [4]. Frequently employed techniques are derived from laboratory setups for the investigation of planar samples. However, artists purposefully create paintings c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 136–144, 2021. https://doi.org/10.1007/978-3-030-58380-4_17

Design and Analysis of Cable-Driven Parallel Robot CaRISA

137

Fig. 1. CAD rendering of proposed cable robot CaRISA depicted scanning Vermeer’s Girl with a Pearl Earring (picture CC-BY-SA 3.0 Unported License).

with topographical surface to provide a level of plasticity and depth. This is particularly interesting for painted marbles from Roman or Greek antiquity [2,7] where correction for variation in the surface was made in post processing. However, not all image distortion is recoverable and may more easily be avoided when capturing an image. Common devices are handheld, or tripod-mounted or table-mounted. Handheld operation limits the analysis to isolated spots and, consequently, does not allow for acquisition of representative data. Tripod-mounted devices can be positioned more freely and can, to some extent, account for surface curvature, yet their scanning range is only a few hundred square centimeters [6]. Table-mounted devices allow for scanning large areas, and, in some cases, can also follow the surface to keep a constant distance, however, the measurement head cannot be panned or tilted [17]. Panning and tilting the view is needed to uncover shaded areas and to keep the plane of focus aligned. All operated systems provide for serial motion of the imaging device, resulting in terminal positioning inaccuracy due to error accumulation of each axis. Using a parallel kinematic manipulator can remedy this disadvantage since the terminal accuracy is not determined by only a single axis’ error. However, they feature a workspace much smaller than their footprint due to limited axis stroke. Replacing the m rigid link actuators with elastic and flexible cables can drastically increase the workspace of the parallel robot yet compromises accuracy and stiffness. A cable-driven parallel robot (cable robot) nevertheless seems appropriate for this type of application due to its large workspace and scalable design. To the best of the authors’ knowledge, only two types of cable robots have been developed for carrying an imaging device. The most well-known system is SkyCam by Cone et al. [5], displacing a camera for aerial view of sports matches. Gosselin et al.developed a cable robot for capturing the appearance of objects using 6-degree of freedom (DoF) with a 1-DoF gravity-powered passive

138

P. Tempel et al.

Fig. 2. Schematic drawings of (a) suspended, (b) fully constrained, and (c) protruding designs.

mechanism end effector displacing a high resolution camera on a hemisphere centered around the object [8]. Practicality of cable robots for scanning art has yet to be investigated. In this contribution, we present a Cable Robot for Inspecting and Scanning Artwork (CaRISA) (see Fig. 1) of non-planar, spatial objects alike to paintings and frescos. Our robot is designed to be used in situ i.e., it is deployed on-site of the artwork and arranges itself around the piece of art. Our resulting robot is novel in that we are providing the application-optimal geometric design for the task of scanning artwork in the general realm of paintings. This paper is structured as follows: Sect. 2 presents the application and hardware requirements from which we infer the cable robot properties and archetype design in Sect. 3. Succeeding is the workspace analysis in Sect. 4 showing the used kinematics model as well as the obtained application-optimal final geometry. Sect. 5 summarizes and concludes the paper.

2

Application Requirements and Specification

We assume the artwork to be a three-dimensional surface over which the imaging device is moved subsequently capturing the whole piece. The positioning device must allow for keeping the optical axis aligned with the scanning area normal. This results in a 5-DoF workspace: 3 translational DoF and 2 rotational DoF for panning and tilting. The spatial extent of the workspace clearly depends on painting size and focal range of the imaging device. We assume it to be (80 × 60) cm in width and height, the size also covered by established instruments. Panning and tilting must be possible for the range of (−35 to 35)◦ , similar to existing devices with only tilting ability. CaRISA is equipped with a Dino-Lite Digital Microscope camera offering imaging in visible and invisible light (UV and IR). The camera is attached to a DinoLite WF-10 WiFi Streamer, providing power over USB and wireless data transmission. The cable robot operates by capturing an image when at rest, thus no live data feed is needed. With a focal distance ranging from 22 mm to 48 mm, the camera has a maximum field of view of 70◦ . The camera’s CMOS chip features (1280 × 1024) px giving an image pixel size of 10 µm to 15 µm.

Design and Analysis of Cable-Driven Parallel Robot CaRISA

139

In situ scanning of the art object implies it be left at its location of installation and the positioning and imaging device being fitted around. As such, the device must be mobile, transportable, adjustable, and allow for scanning outside of its footprint.

3

Structural Design

A cable robot consists of a mobile platform with n DoF driven by m cables. Verhoeven et al. [18] categorize them by their redundancy r = m − n as i) underactuated or incompletely restrained when m ≤ n, ii) completely restrained when m = n + 1, and iii) overactuated or redundantly restrained when m > n + 1. Either category can be operated either suspended i.e., cranelike (CoGiRo [9], Fig. 2(a)), or (fully) constrained (SEGESTA [11], Fig. 2(b)). While crane-like cable robots are unable to withstand all external wrenches and some DoF may be uncontrollable, fully constrained systems can balance arbitrary wrenches and have all DoF controllable. Inherent to most cable robots is a small orientation workspace due to cable-cable collision or infeasibility of attaining static equilibrium. CoGiRo’s special design allows for large rotation of more than 100◦ around one axis [9]. A third, rarely used archetype is the protruding FALCON7-design by Kawamura et al. [12]. It features a thin frame out of which the platform protrudes (see Fig. 2(c)). The design was intended for highspeed pick-and-place applications requiring high stiffness, achieved by the outof-plane cable connections. Besides allowing for a large orientation workspace, this also results in the workspace being located outside of the frame. Given the limitations of a workspace-enclosing design or a crane-like design, choosing the FALCON7-design implies itself.

Fig. 3. Kinematic robot model with world coordinate system . platform coordinate system

and local

140

4

P. Tempel et al.

Workspace Analysis

The tool center point (TCP) must reach every point within a cuboid of dimensions (80 × 10 × 60) cm, the depth of 10 cm chosen to allow also using other imaging devices in a later stage. 4.1

Kinematics Model

We assume the standard kinematic model (cf. Fig. 3) as originally presented by Albus et al. [1], where the mobile platform is positioned at r with orienˆ y is of no tation RP = RPˆ x (α) RPˆ z (γ), pan and tilt; rotation about focal axis P interest. A vector loop is drawn from the i-th frame anchor Ai at ai and attached to the ith platform anchor Bi at bi w.r.t.the platform coordinate system . This yields the cable length from Li = li  = ai − (r + RP bi ) . Static equilibrium of the cable robot can be achieved by balancing all m cable forces fi with the external wrench w such that it reads ⎡ ⎤   f1 u1 . . . um ⎢ .. ⎥ A f + w = + w = 0, (1) b 1 × u1 . . . b m × um ⎣ . ⎦ fm with the cable direction vector ui = li /Li . Above equation describes the structure equation with the structure matrix A [10], and at the core of determining the wrench-feasible workspace (WFW). Even though pulleys are used for guiding the cables, we neglect them in the analysis since their influence on the workspace space is negligibly small for the robot dimensions (see [13] for a discussion). Table 1. Final geometric positions of each cable frame anchor ai and platform anchor bi . Cable Frame anchor aix /m aiy /m

aiz /m

Platform anchor bix /m biy /m biz /m

−1.250 1.250 1.250 −1.250 −1.250 1.250 1.250 −1.250

1.250 1.250 1.250 1.250 −0.750 −0.750 −0.750 −0.750

−0.075 0.075 0.075 −0.075 −0.075 0.075 0.075 −0.075

1 2 3 4 5 6 7 8

0.100 0.100 −0.100 −0.100 0.100 0.100 −0.100 −0.100

0.075 0.075 0.775 0.775 0.050 0.050 0.800 0.800

−0.075 −0.075 −0.100 −0.100 0.075 0.075 0.100 0.100

Design and Analysis of Cable-Driven Parallel Robot CaRISA

4.2

141

Workspace Archetype and Criterion

The TCP shall be positioned in the spatial extent of the cuboid in front of the painting (cf. Sect. 2) such that any orientation within the set of possible rotations can be attained. This total orientation workspace

is the set of poses which contains all positions r where all orientations RP are in a given set of orientations . Pose validity is checked based on wrench-feasibility i.e., checking if there exists a force distribution f ∼ ∈ [f − , f + ] = [10N, 150N] such that Eq. (1) holds true. Since the platform must not touch the artwork,  the external wrench w = mP · [g, (RP  × g) ] comprises only gravitational forces and torques, with total platform mass mP = 5kg, direction of gravity g = [0, 0, −g], and the center of gravity (CoG)  in platform coordinates. We solve Eq. (1) using the Improved Closed-Form by Pott [14], which solves f ∼ = f − A+ (w + A f ) , where f = (f + + f − )/2. If no force distribution can be found, the pose is not within the WFW. 4.3

Application-Optimal Workspace

Our design goal is to find a set of dimensions for frame and mobile platform such that CaRISA covers the whole painting area for all focal distances. As any arbitrarily large configuration will cover the volume of interest, we search for those satisfying additional constraints on the platform shape and maximum frame dimensions. Through gradient-based methods we find a locally optimal solution to the non-linear constrained optimization problem (see methods in [15]). The final frame and platform dimensions are given in Table 1, which shows cable attachment in a crosswise sequence, yet cable-cable collision does not occur in the volume of interest. Figure 4 shows the workspace for multiple orientations of the mobile platform to account for the desired range of pan and tilt. All figures show the desired workspace in light green, the platform in dark gray, the cables in red, and the obtainable translational workspace for each rotation value. ˆz close to zero covers ˆx or e The translational workspace for rotations about e almost 90% of the robot’s frame area. For maximum angles of pan and tilt, coverage reduces to approximately 45%. In either case, however, the desired volume is ˆx i.e., the still within the translational WFW. For large positive rotation about e camera pointing downwards, and due to the TCP position not coinciding with the platform’s CoG, the WFW is of much smaller size compared to negative ˆx i.e., with the camera pointing upwards. We conclude possirotations about e bility of finding an application requirements-satisfying geometry for scanning of artwork. From a theoretical point of view, accuracy and repeatability of the device are in the required range influenced only by the drive encoder and gear ratio.

142

P. Tempel et al.

Fig. 4. Translation wrench-feasible workspace for multiple orientations of the platform.

However, well-known physical challenges of cable robots of cable elasticity and flexibility do not allow for making a blanket statement on the obtainable terminal accuracy. As of writing, the physical prototype is being constructed. We are already developing further improvements to the system such as additional cable force sensors and absolute cable length measurement devices to combat cable elasticity and induced accuracy impairment. We expect the final workspace size to be smaller to the analytical result for that 1) friction of pulleys and bearings is neglected but will inevitably impair the transmitted drive torque/force, and 2) the final assembly will not share the exact same geometry as determined analytically due to manufacturing inaccuracies and possible adjustments to the mechanical design. Despite that, the actual physical workspace is expected to be marginally smaller than the analytical workspace, which is why a theoretically larger workspace is generally preferred during design–see also [15] for a critical review of this fact.

Design and Analysis of Cable-Driven Parallel Robot CaRISA

5

143

Conclusions

We present the 5-DoF Cable Robot for Inspecting and Scanning Artwork (CaRISA) of cultural significance with a large translation and orientation workspace. Cable robots deems themselves a good choice due to their lightweight and modular design and their large workspace. Conventional cable robot designs are limited in their orientation workspace, which is why we choose the FALCON7-design. CaRISA’s geometry is optimized for the application requirements of achieving a wrench-feasible workspace with panning and tilting of the imaging of 70◦ each, respectively. Acknowledgment. The authors would like to thank Joris Dik (Delft University of Technology, Department of Materials Science and Engineering) for fruitful discussions and his contributions to the research. We would also like to thank our colleagues at the Rijksmuseum Amsterdam, with whom we have developed the concept of CaRISA in close collaboration. The results shown in this contribution were in parts generated using WireX, an open source software for analysis and design of cable robots [16].

References 1. Albus, J.S., Bostelman, R.V., Dagalakis, N.G.: The NIST ROBOCRANE: originally published as “The NIST SPIDER, A Robot Crane”. J. Res. Natl. Inst. Stand. Technol. 97(3), 373–385 (1992) 2. Alfeld, M., Mulliez, M., Martinez, P., Cain, K., Jockey, P., Walter, P.: The eye of the medusa: XRF Imaging reveals unknown traces of antique polychromy. Anal. Chem. 89(3), 1493–1500 (2017) 3. Alfeld, M., Siddons, D.P., Janssens, K., Dik, J., Woll, A., Kirkham, R., Wetering, E.: Visualizing the 17th century underpainting in Portrait of an Old Man by Rembrandt van Rijn using synchrotron-based scanning macro-XRF. Appl. Phys. A, Mater. Sci. Process. 111(1), 157–164 (2013) 4. Alfeld, M., de Viguerie, L.: Recent developments in spectroscopic imaging techniques for historical paintings: a review. Spectrochim. Acta, Part B 136, 81–105 (2017) 5. Cone, L.L.: Skycam, an aerial robotic camera system. Byte Mag. 10, 122–132 (1985) 6. Galli, A., Caccia, M., Alberti, R., Bonizzoni, L., Aresi, N., Frizzi, T., Bombelli, L., Gironda, M., Martini, M.: Discovering the material palette of the artist: a pXRF stratigraphic study of the giotto panel “God the Father with Angels”. X-Ray Spectrom. 46(5), 435–441 (2017) 7. Geil, E.C., Thorne, R.E.: Correcting for surface topography in X-ray fluorescence imaging. J. Synchrot. Radiat. 21(6), 1358–1363 (2014) 8. Gosselin, C.M., Bouchard, S.: A gravity-powered mechanism for extending the workspace of a cable-driven parallel mechanism: application to the appearance modelling of objects. Int. J. Autom. Technol. 4(4), 372–379 (2010) 9. Gouttefarde, M., Merlet, J.P., Daney, D.: Wrench-feasible workspace of parallel cable-driven mechanisms. In: 2007 IEEE/RAS International Conference on Robotics and Automation (ICRA 2007), pp. 1492–1497 (2007)

144

P. Tempel et al.

10. Higuchi, T., Ming, A., Jiang-yu, J.: Application of multi-dimensional wire cranes in construction. In: The 5th International Symposium on Robotics in Construction, pp. 661–668 (1988) 11. Hiller, M., Fang, S., Mielczarek, S., Verhoeven, R., Franitza, D.: Design, analysis and realization of tendon-based parallel manipulators. Mech. Mach. Theory 40(4), 429–445 (2005) 12. Kawamura, S., Choe, W., Tanaka, S., Pandian, S.R.: Development of an ultrahigh speed robot FALCON using wire drive system. In: 1995 IEEE/RAS International Conference on Robotics and Automation (ICRA 1995), pp. 215–220. IEEE (1995) 13. Pott, A.: Influence of pulley kinematics on cable-driven parallel robots. In: Lenarˇciˇc, J., Husty, M.L. (eds.) Latest Advances in Robot Kinematics, pp. 197–204. Springer, Dordrecht (2012) 14. Pott, A.: An improved force distribution algorithm for over-constrained cabledriven parallel robots. In: Thomas, F., P´erez-Gracia, A. (eds.) Computational Kinematics, Mechanisms and Machine Science, vol. 15, pp. 139–146. Springer, Dordrecht (2014) 15. Pott, A.: Cable-Driven Parallel Robots: Theory and Application, Springer Tracts in Advanced Robotics, vol. 120. Springer International Publishing, Cham (2018) 16. Pott, A.: WireX: An Open Source Initiative Scientific Software for Analysis and Design of Cable-Driven Parallel Robots (2019) 17. Romano, F.P., Caliri, C., Nicotra, P., di Martino, S., Pappalardo, L., Rizzo, F., Santos, H.C.: Real-time elemental imaging of large dimension paintings with a novel mobile macro X-ray fluorescence (MA-XRF) scanning technique. J. Anal. At. Spectrom. 32(4), 773–781 (2017) 18. Verhoeven, R., Hiller, M., Tadokoro, S.: Workspace, stiffness, singularities and classification of tendon-driven stewart platforms. In: Lenarˇciˇc, J., Husty, M.L. (eds.) Advances in Robot Kinematics, pp. 105–114. Springer, Dordrecht (1998)

A Low Cost Introductory Platform for Advanced Robotic Control Bin Wei(B) Algoma University, Sault Ste. Marie, ON P6A 2G4, Canada [email protected]

Abstract. This study seeks to outline the various factors that culminated in the creation of an educational robotics platform geared towards student audiences. Throughout the study, justifications will be provided for all design decisions, comparisons will be made to alternative commercially available products, and mathematical models for robot performance will be discussed. Keywords: Robotics · Controls · Python · Educational robot · Arduino microcontroller · Trajectory planning

1 Background Around the globe, companies are becoming increasingly dependent on automated manufacturing systems in order to maintain production targets. This makes it of utmost importance that the next generation of our workforce have a basic understanding of robotic control concepts and software development. We chose to design and build a low-cost educational robot geared towards students that may be used in schools to teach these concepts before their exposure to similar technologies in the workforce. There are quite a few papers focusing on the educational robots design (Baxter et al. 2017; Beer et al. 1999; Fridin et al. 2014; Kennedy et al. 2015; Kanda et al. 2007; Nourbakhsh 2000). The main difference between this study and others is that a low cost introductory platform for advanced robotic control is designed here.

2 Design Process In order for educators to widely adopt our educational robot, the designed product needed to be inexpensive, easy to use, and functionally engaging for the students using it. Together, these three objectives drove many of the design considerations for our robotic arm.

© CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 145–153, 2021. https://doi.org/10.1007/978-3-030-58380-4_18

146

B. Wei

2.1 3D Printed Arm Segments For students who are relatively new to the product manufacturing space, it was important that the design the robotic arm in such a way that it was easily manufacturable on a mass scale without the large startup costs normally associated with having appropriate tooling designed to support new product manufacturing. These days, 3D printing technology has become increasingly popular with equipment and materials available for inexpensive costs. It is decided that in order to meet the cost objectives, we would ensure that all arm segments could be modelled in CAD and easily 3D printed using ABS or PLA filament. This methodology would also allow for students to 3D print their own arm segments and end effector designs if equipped with appropriate tools. 2.2 Servo Controlled Revolute Joints With the decision made to manufacture arm segments using 3D printing technology, we next needed to decide how to facilitate movement between each arm segment. By doing research into available solutions, we settled on using off-the-shelf nine gram servo motors for two reasons. First, these motors are extremely low cost and are readily available online in appropriate quantities that can be scaled to support production as needed. Second, servo motors have built-in position sensors and an on-board logic controller. This means that from a software standpoint, we only would have to provide an angle while the servo controllers would take care of moving to that position. This would avoid needing to design and implement a separate control loop involving the use of external position sensors. As a bonus, CAD models of the nine gram servos were readily available in order to generate models of each arm segment to be 3D printed. 2.3 Arduino Microcontroller The final major design decision made was determining which micro-controller we would use in order to facilitate control of each servo motor. We needed to ensure that the selected microcontroller would support at least 5 PWM outputs, could easily be programmed, and could be communicated with using a standard serial protocol. Here the Arduino Nano microcontroller is used for this purpose. 2.4 Design Simulation This simulation was generated using a SolidWorks 2017 motion study which was then passed into SolidWorks Visualization for final rendering, as shown in Fig. 1.

A Low Cost Introductory Platform for Advanced Robotic Control

147

Fig. 1. 3D mode of the robot

3 Comparison 3.1 VEX Robotics VEX Robotics is a robotic design platform aimed at teaching children robotic design principles. The robots are be built out of metal and plastic parts, then programmed with a programming language that is a derivative of C, known as RobotC. VEX products are sold online and in stores at different cost levels depending on product version and package. VEX’s new V5 products sell for upwards of $700, and their legacy products sell for upwards of $500. While these products provide an in depth view of robotic design principles, they have a steep learning curve, and a massive upfront cost. 3.2 OWI Robotic Arm Edge OWI535 OWL Robotics as another company focused on educating children about robotic systems. The OWL robotic arm (Cerkala 2013) is a simple robotic arm designed to be built and used with a simple manual controller. It allows users to pick up, move, and place down object through the use of a manually operated remote. While manual control provides users with the understanding of how joints work, and how they affect the system, there are no options for controlling the system programmatically. This basic robotic arm is marketed for $75 which is a decently low cost point. Without the ability to program paths or angles into the robotic arm, major robotic design principles, such as path planning algorithms, and other mathematical principles cannot be taught well on the platform. 3.3 Lego Mindstorms EV3 Lego Mindstorms allows for the development of simple robotic systems though custom Lego pieces and simple controllers. Programming its EV3 products is primarily done through a visual programming language (drag and drop components) called NXT-G, but controllers can optionally be programmed with more advanced languages such as RobotC. Entry level Mindstorms products sell for upwards of $400. A major drawback for these products is that the recommended and supported NXT-G does not allow for a complicated control algorithm for the robot’s motors. The language is designed with general use in mind, and does not focus on robotic arm designs. The cost is also a major issue for many users.

148

B. Wei

3.4 Our Design Our design targets and resolves these issues to provide users with a more in depth experience with robotic arm designs. It allows users to modify specific joint conditions and experiment with different path planning algorithms, all at a fraction of the cost of other the systems that we compared, as illustrated in Table 1. Table 1. Comparisons among different groups Our Design

VEX robotics

OWI robotic arm edge OWI535

Lego Mindstorms

Cost

$30

$700

$75

$400

Feedback

Yes

Yes

No

Yes

Computer control

Yes

Yes

No

Yes

Assembly

Easy

Complicated

Complicated

Easy

Ease of Use

Easy

Difficult

Easy

Easy

4 Comparison The following calculations were preformed to emphasize how changes in each joint variable (as a function of time) drastically vary the trajectory of the end effector and the joint torques. Using Matlab’s optimal environment for intensive applications of linear algebra, symbolic representations of these desired features can easily be computed and plotted for visual illustration. 4.1 Function: DH2JntSpc(DH; q; q t; m; I) Inputs: symbolic DH parameters with n number of rows (joints) as cell object, symbolic variable vector of n length, symbolic function vector of n length, symbolic mass vector of n length, symbolic moment of inertia vector of n length. Outputs: symbolic transformation matrices as a cell object of n entries, symbolic transformation matrix with respect to the base, symbolic Lagrange components of the joint torques (M, V, and G), symbolic joint torque vector.

4.2 Initialization of Symbolic Matrices and Other Entities These are necessary for faster execution time.

A Low Cost Introductory Platform for Advanced Robotic Control

149

4.3 DH Parameters → Joint Space for N Joints 4.3.1 Joint DH parameters → Transformation matrix i−1 Ti Each row of DH parameters is converted into a respective transformation matrix using the general transformation formula.

4.3.2

I−1 T i

→ 0 Ti (0 Tn at End Effector) and Joint Jacobian Ji

The reference to the base uses Euler representation, where the transformation is based ∧

off the previous configuration (post multiply). The rotation vector 0 zi (the 3rd column, first 3 rows of 0 Ti ) and the position vector 0 xpi (the 4th column, first 3 rows of 0 Ti ), are extracted for the use of the Jacobian Ji . Note that since the specific application has no ∧

prismatic joints, 0 zi is valid for all joints. In the future, this would be changed to detect the presence of a translational variable in the 4th column of i−1 Ti , which would make the last 3 rows of the corresponding Ji column = [0; 0; 0].

150

B. Wei

4.3.3 Ji → Mass Matrix M and Gravitational Force Vector G For each iteration, the mass/inertia of the new joint is added to the mass matrix based on Ji . The same is done with the gravitational term, which only uses the linear portion of Ji . Note that the gravity vector is hard set for the specific application. In the future, this will be changed to determine the axis of gravity based on the location of a joint variable in i−1 Ti .

4.3.4 M → Centrifugal and Coriolis Vector V The derivative of M with respect to time is taken for the centrifugal term, and the derivative of M with respect to each joint variable is taken at the corresponding row for the Coriolis term.

4.3.5 M, V, G → Joint Torques τ Since the joint variables are real continuous functions with respect to time, symbolic functions replace the symbolic variables. This ensures the proper output when substituting double functions into the returned matrices of 0 Tn , M, V, G and τ .

5 Outputs The following outputs (Figs. 2, 3, 4, and 5) are generated from the default inputs, where all joint variables are assigned to π2 sin(t). The joint functions can be changed during run time using the command line.

A Low Cost Introductory Platform for Advanced Robotic Control

151

Fig. 2. 3D position of end effector

Fig. 3. Variations in Cartesian coordinates over time

Fig. 4. Joint τ as time progresses

Fig. 5. Joint τ with respect to corresponding joint variables

6 Graphical Control Interface When designing the control interface, we decided to utilize a combination of tools and libraries that would make the development process fast and efficient. Thus we decided on the following technology stack: High performance QT 5.1 C++ graphical library, Python 3.6, PySide2 to enable Python 3.6 bindings to QT 5.1, Matplotlib for efficient graphs, SciPy for algorithms, NumPy for mathematical operations. There are two main parts in the control interface; graphics and control. The graphical layout and graphics interaction code is relatively straight forward, thanks to the powerful Qt 5.1 library. The control section dictates how the joint setpoints are translated into smooth curves for the servos. The control code has multiple ways of generating paths. Initial algorithms used were simply interpolation algorithms that included: Linear interpolation, Cubic interpolation, Radial Basis Function (RBF) interpolation, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation. After these interpolation algorithms were implemented, we then decided to include the linear interpolation with parabolic blend algorithm (linear parablend). Such an algorithm is not trivial to implement, but it was finished after a lot

152

B. Wei

of math was translated into Python code. The control interface to the Arduino microcontroller is implemented as a separate thread, so that the graphical interface can be updated in real time while serial commands are fed to the physical arm. 6.1 Time Optimization After the linear parablend algorithm was finished, we then realized that given the physical servo parameters (maximum acceleration and maximum velocity), the final path could be time-optimized by shrinking the time between points until a physical servo parameter is violated. This was implemented in an iterative way using Newton’s numerical analysis method. This method attempts to decrease the time between points by ti , regenerating the path and checking if it is still valid. If it is valid, decrease by ti and check again. If it is not valid ti is divided by 2 and the path is regenerated and checked again. This optimization process repeats until ti 0 (integration step), sk0 ← [1, 1, ..., 1]T , tk > 0, tf > 0 while |Eq |>1 , |Eq˙ |>2 , |Eq¨|>3 , |Eqs |>4 , |Eq˙s |>5 , |Eq¨s |>6 , i ≤ maxi do ... ... ˙ q ¨ , qs , q˙ s , q ¨ s ]i = RK4(tint , sk , h, ( q , q s ))  Numerical integration with [q, q, Runge-Kutta 4-th order of Eqs. (4) ⎧ ∗ E (s ) = q(t , s ) − q ⎪ q k f k t f ⎪ ⎪ ⎪ ⎪Eq˙ (sk ) = q(t ˙ f , sk ) − q˙ ∗tf ⎪ ⎪ ⎪ ⎨E (s ) = q ¨ (tf , sk ) − q ¨ ∗tf q¨ k  BVP errors (* means desired value) [E(tf , sk )]i = ⎪Eqs (sk ) = qs (tf , sk ) − q∗stf ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎪Eq˙s (sk ) = q˙ s (tf , sk ) − q˙ stf ⎪ ⎩ ¨ s (tf , sk ) − q ¨ ∗stf Eq¨s (sk ) = q ∂E(tf , sk )  Gradients [J]i = ∂sk [Δsk ]i = LMUpdate(sk , J)  Levenberg-Marquardt update based on [13] ˙ k ), q ¨ (tk ), [sk ]i+1 = [sk ]i + [Δsk ]i  Update of parameter vector sk = [q(tk ), q(t ¨ s (tk )]T qs (tk ), q˙ s (tk ), q end while ˙ int , sk ), q ¨ (tint , sk ), qs (tint , sk ), q˙ s (tint , sk ), q ¨ s (tint , sk ) Output: sk , q(tint , sk ), q(t

be specified. Thus, since we seek to apply the BVP formulation to constrain up to the accelerations of the pick-and-place points, it is required to have higher-order dynamic equations. This is due to the fact that the order of the system in (1) allows to constrain up the velocities of the pick-and-place task. Therefore, we timed differentiate the expressions in (1) to obtain the jerk equations as follows: ... ˙ q + c˙ + K(q˙ − q˙ s ) + f˙a , τ˙ vss = Js ... q s − K(q˙ − q˙ s ) + f˙s (3) τ˙ = M q + M¨ where the sign functions from the active joint friction terms in fa and fs , in order to obtain f˙a and f˙s , respectively, are approximated to be time differentiable. Then, the BVP formulation applied to find the combined spring and robot coordinates qs and q, respectively, seeks to solve the coupled robot-plusVSS system (3) for τ˙ =0 and τ˙ vss =0, i.e.: ... ˙ q + c˙ + K(q˙ − q˙ s ) + f˙a ), ... q = −M−1 (M¨ q s = J−1 ˙ − q˙ s ) − f˙s ) (4) s (K(q with the boundary conditions defined as the desired positions, velocities and ˙ f ) = q˙ ∗tf , q ¨ (tf ) = q ¨ ∗tf , and accelerations at the braking phase by q(tf ) = q∗tf , q(t the desired final VSS positions, velocities and accelerations by qs (tf ) = q∗stf , ¨ s (tf ) = q ¨ ∗stf by formulating the nonlinear BVP defined q˙ s (tf ) = q˙ ∗stf , and q in Algorithm 1. It should be noted that from the aforementioned algorithm, to compute the BVP error vectors grouped in E(tf , sk ), it is necessary for M to be numerically invertible, i.e. out of Type 2 singularity [14]. Thus, in the integration

218

R. Balderas Hill et al.

Table 1. RMS values of torques for nominal and VSS actuation modes.

Seg

Nominal

Using VSS in parallel

Gain

RMS. Torques

RMS. Torques

torques energy

Time(s) τ RMS (Nm)

τ RMS (Nm)

τ vssRMS (Nm)

Gain

%

%

Fig. 2a 0.51

[5.65, 3.41, 3.85] [0.43, 0.49, 0.58] [0.48, 0.56, 0.59] 75

72

Fig. 3a 0.52

[5.35, 6.53, 3.89] [0.47, 0.51, 0.60] [0.52, 0.61, 0.65] 77

71

step, an inversion-checking condition representing the proximity to singularity is added to verify the invertibility of the decoupling matrix M. The different elements of the pseudo-code provided in Algorithm 1 are defined as follows: maxi is the maximum number of iterations i, 1...6 represent the BVP ¨ , q, ˙ q and q ¨ s , q˙ s , qs are obtained from numerical integration errors q ... thresholds, ... of q and q s , respectively, and E(tf , sk ) groups the BVP error vectors to be minimized. The parameter vector sk , used as decision variable vector to enforce the convergence of the BVP error vectors (See Algorithm 1), is defined as a vector of robot positions, velocities and accelerations, and of VSS positions, velocities and accelerations, at the time instant tk representing the time when the robot braking phase starts, and occurring between t0 and tf of the BVP: ˙ k ), q ¨ (tk ), qs (tk ), q˙ s (tk ), q ¨ s (tk )]T as shown in Fig. 1b, tk ∈ [t0 , tf ] sk = [q(tk ), q(t (only one tk is defined since we have a squared BVP problem). In order to define the braking instant tk , we evaluate the dynamic equations grouped in (1) for different values of tk on the motion segment valued from t0 to tk . This is done in order to choose the one which demands less input torques (τ , τ vss ), similar to what it is done in [15] in which the total motion segment from t0 to tf is divided to solve the BVP. Finally, RK4 represents the numerical integration of the expressions in (4) by using a Runge-Kutta method, and LMUpdate is a function representing the Levenberg-Marquardt updating law based on [13].

4

Simulation Results

For validating the proposed approach, the numerical values of the dynamic parameters for the Delta robot were defined according to a real 3-DOF Delta robot from MG-Tech [11], with the following values: (i) proximal links 2 mass and inertia, respectively: 0.82 kg and 0.017 kg.m2 about its center of mass, located at a distance of 0.115 m from the rotation center of the motors qi ; (ii) parallelogram 3 mass and inertia, respectively: 0.68 kg and 0.024 kg.m2 about its center of mass, located at the middle of the parallelogram; (iii) platform mass 4 : 0.72 kg; (iv) Coulomb friction terms of the actuated joints: 0.45 Nm. The numerical values of the dynamic parameters associated to the VSS placed in parallel to the actuated links were defined, respectively: inertias of the couplings between the motors and the VSS, J1 = J2 = J3 = 0.0051 kg.m2 , and stiffness constants, k1 = k2 = k3 = 4.95 Nm/rad. It is worth to mention that

Minimizing the Energy Consumption of a Delta Robot

219

the rotor inertias, J1 , J2 and J3 were defined according to the motor specifications in [8], whereas the spring constants of the VSS, k1 , k2 and k3 were defined according to the maximum allowable spring deformation, computed by setting the maximum input torques for the robot actuators based on the specifications in [8], and testing for several typical high-speed trajectories. To validate the theoretical formulations, two actuation modes are considered in the numerical simulations: (i) nominal actuation, which consists of the Delta robot without springs, and with a classical fifth-degree polynomial trajectory, and (ii) by using VSS in parallel, and with the proposed formulation from Algorithm 1. Additionally, the BVP approach is tested in two scenarios according to the sequence defined in Fig. 2a and Fig. 3a, respectively. Based on the RMS values of the torques in Table 1, it can be seen that the reduction in the full actuation chain by using VSS can reach 75% and 77% with respect to the nominal actuation mode, respectively for each task. In terms of energy reduction, from evaluating the energetic model (2), in Fig. 2b, and Fig. 3b, it can be seen that by using VSS, it is possible to reduce the energetic losses up to 71%, and 72%, respectively, in the full actuation chain (robot and VSS) with respect to the nominal case. It is worth noticing that here only two scenarios in Fig. 2a

(a) Pick-and-place sequence.

(b) Energetic losses.

Fig. 2. On the left, the sequence is defined by: G → H (time: 0.12 s), H → I (time: 0.06 s), I → J (time: 0.14 s), J → K (time: 0.08 s), K → L (time: 0.11 s). On the right, the energetic losses for the two actuation cases.

(a) Pick-and-place sequence.

(b) Energetic losses.

Fig. 3. On the left, the sequence defined by: G → H (time: 0.12 s), H → I (time: 0.06 s), I → J (time: 0.1 s), J → K (time: 0.06 s), K → L (time: 0.1 s), L → M (time: 0.08 s). On the right, the energetic losses for the two actuation cases.

220

R. Balderas Hill et al.

and Fig. 3a are presented. Nevertheless, other simulations, not presented here due to page limitations, gave similar results in terms of energy reduction.

5

Conclusions and Future Works

This paper proposes a strategy for generating offline optimal motions in order to increase the energy efficiency of a Delta robot by placing variable stiffness springs (VSS) in parallel to the robot active joints. The VSS in parallel configuration was combined with a BVP to find a combined optimal motion of robot and VSS joints. Simulations led to a considerably increase of energy efficiency with multiple-point trajectories on a Delta robot for two cases: nominal (without VSS) and by using VSS. Results show that by using VSS, the energy reduction for the full actuation chain on the Delta robot, i.e. robot-plus-VSS, can reach up to 70% with respect to the nominal case. The future works include the development of a strategy that takes into account the uncertainties on the dynamic parameters as here we assume the numerical integration of a perfect model. In addition to that, future works include as well the experimental validation of our approach. Acknowledgment. This work was conducted with the support of the Mexican Council for Science and Technology (CONACYT), which had fund a PhD grant.

References 1. Brossog, M., Bornschlegl, M., Franke, J.: Reducing the energy consumption of industrial robots in manufacturing systems. Int. J. Adv. Manuf. Technol. 78(5–8), 1315–1328 (2015) 2. Kim, Y.J.: Anthropomorphic low-inertia high-stiffness manipulator for high-speed safe interaction. IEEE Trans. Robot. 33(6), 1358–1374 (2017) 3. Briot, S., Goldsztejn, A.: Topology optimization of industrial robots: application to a five-bar mechanism. Mech. Mach. Theory 120, 30–56 (2018) 4. Uemura, M., Kawamura, S.: Resonance-based motion control method for multijoint robot through combining stiffness adaptation and iterative learning control. In: IEEE International Conference on Robotics and Automation, Kobe, Japan, pp. 1543–1548 (2009) 5. Barreto, J.P., Sch¨ oler, F.J.-F., Corves, B.: The concept of natural motion for pick and place operations. In: New Advances in Mechanisms, Mechanical Transmissions and Robotics. Springer, pp. 89–98 (2017) 6. Barreto, J.P., Corves, B.: Resonant delta robot for pick-and-place operations. In: Advances in Mechanism and Machine Science, pp. 2309–2318 (2019) 7. Goya, H., Matsusaka, K., Uemura, M., Nishioka, Y., Kawamura, S.: Realization of high-energy efficient pick-and-place tasks of scara robots by resonance. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2012), Vilamoura, Algarve (Portugal), October 2012, pp. 2730-2735 (2012) 8. Balderas Hill, R., Briot, S., Chriette, A., Martinet, P.: Increasing energy efficiency of high-speed parallel robots by using variable stiffness springs and optimal motion generation. In: Proceedings of the ASME 2018 IDETC/CIE 2018, pp. V05BT07A011 10 pages, DETC2018-85090, Quebec City, Canada (2018)

Minimizing the Energy Consumption of a Delta Robot

221

9. Balderas Hill, R., Briot, S., Chriette, A., Martinet, P.: Minimizing input torques of a high-speed five-bar mechanism by using variable stiffness springs. In: Proceedings of the 22nd CISM-IFToMM Symposium on Robot Design, Dynamics, and Control (ROMANSY 2018), Rennes, France, 25–28 June 2018, pp. 61–68 (2018) 10. Hollander, K., Sugar, T.: Concepts for compliant actuation in wearable robotic systems. In: Proceedings of the US-Korea Conference on Science, Technology and Entrepreneurship (UKC 2004), vol. 128, pp. 644–650 (2004) 11. MG-Tech. https://www.mg-tech.fr/ 12. Briot, S., Khalil, W.: Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements. Springer, Cham (2015). ISBN 9783319197883 13. Mor´e, J.J.: The Levenberg-Marquardt algorithm: Implementation and theory. In: Watson, G.A. (ed.) Numerical Analysis. Lecture Notes in Mathematics, vol. 630. Springer, Heidelberg (1978) 14. Gosselin, C., Angeles, J.: Singularity analysis of closed-loop kinematic chains. IEEE Trans. Rob. Aut. 6(3), 281–290 (1990) 15. Balderas Hill, R., Briot, S., Chriette, A., Martinet, P.: Exploiting natural dynamics in order to increase the feasible static-wrench workspace of robots. In: Proceedings of the 15th IFToMM World Congress, 30 June–4 July 2019, Krakow, Poland (2019)

Preliminary Design and Modeling of a Robot for Pipe Navigation with a Novel Wheel-Leg Architecture Carl A. Nelson(B) University of Nebraska-Lincoln, Lincoln, NE 68588, USA [email protected]

Abstract. Robots for pipe inspection have become more prevalent in recent years, and expanding the capabilities of these robots is of interest. Specific needs include accommodating a larger range of pipe sizes without needing customized robots, and navigating sharp bends in pipe networks. This paper presents a new pipe robot architecture with emphasis on a novel multi-link wheel-leg design. Favorable properties of the kinematics and kinetostatics of the robot are illustrated through analysis and simulation, and bounds for feasible component dimensions based on task descriptions are presented. Keywords: Pipe inspection robot · Wheel-leg design · Feasible design space · Kinetostatic analysis

1 Introduction Inspection of pipeline networks represents a vast industry throughout the world. To provide a sense of scope, there are more than 3.8 million km of pipelines in the Unites States energy network alone, and these require regular inspection and maintenance. In addition to external inspection, a common method of internal pipe inspection is to navigate a “pig” or traveling inspection device between ports in the pipeline designed for this purpose. Unfortunately, a large percentage of pipe sections are not suitable for this technique, either due to bends in the pipeline or other factors. Robotic approaches to pipe inspection have become more and more prevalent over the past two decades to fill these needs. Many pipeline robot designs have been proposed, varying in size, locomotion strategy, and capabilities. For example, one research group chose an inchworm-type locomotion with a passive spring in between modules to both conform to bends and allow variation of the inter-segmental distance [1]. Another inchworm-type robot was proposed recently [2] with tensegrity units in between segments to provide active steering. Many robot designers, however, have preferred wheeled or tracked locomotion. At the most basic level, a pipeline robot can just be a simple wheeled rover [3], but this can limit robot capabilities in terms of climbing inclines and controlling tractive force. Therefore, many robots include a means of adjusting the effective diameter of the robot and/or the © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 222–229, 2021. https://doi.org/10.1007/978-3-030-58380-4_27

Preliminary Design and Modeling of a Robot

223

force applied to the pipe wall at the wheel or track. From the point of view of kinematics, there are several “leg” mechanisms which have been extensively used to achieve this size/force adaptation, including the scissors [4, 5], the parallelogram (four-bar linkage) [6, 7], and the slider-crank linkage [7–9]. Other approaches, such as simple pivoted legs [10], prismatically extending legs [11], and four-bar mechanisms [12], have also been used. Many of these robots use springs to passively adjust shape and maintain contact with the pipe wall [1, 4, 5, 7–11]. Conspicuously absent, however, is a leg design which accommodates large variations in pipe diameter. Steering is also a feature that is often under-addressed in these robots, due to either passive pipe-following approaches or inadequate range of steering motion. In this paper, we present a novel robot design for navigating inside pipes with a large range of sizes. The characteristics of the design also lend themselves to passively maintaining good contact distribution at the wheels and navigating difficult bends. The design approach and modeling are laid out in the sections that follow.

2 Overall Robot Design The robot should satisfy the following requirements: • • • •

Passively adjust to small changes in pipe profile Actively adjust to large changes in pipe profile Accommodate profile diameter changes greater than the robot module length Accommodate both gradual and sharp (90°) bends

To satisfy all of these requirements can be challenging. For example, in a larger pipe section, a relatively small robot would have difficulty navigating a sharp bend (or “intersection”) due to discontinuities of wheel contact, and in a narrower pipe section the difficulty might come from impingement of the robot body against the corner of the pipe at such an intersection. To resolve these issues, the chosen architecture uses two identical modules (as in [1, 2]); however, rather than using an inchworm approach, each module has two sets of four independently driven wheel-legs, and a differential joint connects the two modules for steering (Fig. 1). Thus, each of the two modules possesses 9 DOF (8 wheel drives, and one axis of relative motion between modules), not counting the articulation of the wheel-legs. It should be noted that the drive DOFs could be reduced to as few as 1 per module, but this would increase mechanism complexity (since wheel-legs independently passively adjust to facilitate wall contact) and would require slip at the wheels to navigate certain bends, so this is a design choice or tradeoff (i.e., all-wheel-drive considered superior to skid-steering).

224

C. A. Nelson

Fig. 1. Robot architecture (side and front views): W = wheel with drive motor, D = differential.

3 Wheel-Leg Design Because of the mechanisms chosen to adjust to different pipe sizes, previous pipe robots generally are not able to accommodate pipe diameters significantly greater than one robot length (typically a pair of legs approximately half as long as the robot are stowed against the robot body for narrow pipe and deployed outward for large pipe [4–10]). Here we propose to use a modular multi-link leg such that larger diameters can be accommodated by adding leg segments. To simplify the leg design, we replace parallelogram, scissor, or slider-crank linkages with cable-pulley constraints, such that “even” leg segments and “odd” leg segments remain respectively parallel to each other (this results in a lighter and more compact leg). As shown in Fig. 2, each leg segment is very simple, with a single

Fig. 2. Wheel-leg design: (left) basic schematic showing spring-loaded winch, articulated leg, leg extension spring, and wheel; (center) CAD model of partial leg, showing pulleys (red, green) and cable mounting point (yellow); (right) ADAMS model of partial leg showing cable routing (white) and pulleys (pink, sizes exaggerated for illustration); (bottom) routing for (blue and red) cables to enforce equal leg-segment angles.

Preliminary Design and Modeling of a Robot

225

cable wrapping over two pulleys from an anchor point on the preceding link to an anchor point on the following link (this is kinematically equivalent to a parallelogram linkage). Since the relative angular displacement of the leg segment pairs is double that of the first leg segment with respect to the robot body (Fig. 2 left), a single half-size pulley is needed; otherwise all pulleys are the same diameter. A torsion spring at the distal leg segment biases the leg to extend, and accommodates small changes in pipe profile, while a winch with a series spring is used to retract the leg to accommodate large changes in pipe diameter and/or regulate the effective stiffness of the leg-pipe interaction. Drive wheels and motors are located at the distal point of each leg.

4 Modeling and Simulation Whereas the preceding sections outline the robot architecture and general design, the details related to parameter selection (values of length L, width w, and number of leg segments n) are not yet determined. These are tied to the particular requirements of the robot, especially the desired range of pipe sizes [dmin dmax ]. In this section we illustrate the relationships that can be generally applied in adapting this robot design for specific task requirements (see Figs. 3 and 4 for illustration). To achieve a 90° bend, the length of the differential between modules should be at least the minimum module width, to prevent self-collision of the two modules when bent at 90°, as shown in Fig. 3. In this analysis, we adopt a conservative assumption for navigating 90° bends. This assumption is that as a robot module turns a corner, its

Fig. 3. Step-wise process for navigating a sharp bend; active contact interfaces (wheel-leg extended) denoted “C”.

Fig. 4. Diagram illustrating formulation of bend-navigation constraint.

226

C. A. Nelson

end-points travel on straight lines parallel to the center line of the pipe section, in a manner similar to a Scotch-yoke mechanism. Since the “coupler” link of a Scotch-yoke mechanism is closest to impinging at the intersection of the two perpendicular tracks when at 45° (see Fig. 4), we can model the collision-free condition at the smallest pipe diameter as follows. (L/2 + w)(sin45◦ ) < dmin

(1)

To complete a turn and allow the forward module to gain traction and pull the rear module through (see Fig. 3 center), it is necessary that L>w

(2)

In order for the robot to fit inside the smallest desired pipe diameter, we have w < dmin

(3)

w > 2nt

(4)

where the width is constrained by

and t is the thickness of each leg segment (viewed from the side) and n is the number of leg segments per leg; this involves a simplified assumption that the leg segments can collapse to lie flat against each other. Since each leg segment has length at most l = L/2, in order for the legs to reach the pipe wall and establish frictional contact, the number of leg segments is n > dmax /L

(5)

Collectively, these design constraints form a feasible design space in three dimensions (n, w, and L), and each constraint equation partitions the design space (Fig. 5).

Fig. 5. Planar slices of the feasible design space in w, L, and n.

Preliminary Design and Modeling of a Robot

227

We can also model the normal force F that can be generated at the wheel-wall interface. The kinematic relationship between the leg extension h and the angle of a single leg segment θ with respect to the axis of the robot body is h = nlsinθ

(6)

where l is the leg segment length, assumed here to be L/2. We assume a linear torsion spring as shown at the distal leg joint in Fig. 2 (left), which reaches its free state when the leg is fully extended (leg segments align) and is displaced to an angle φ as the leg is folded. Applying the principle of virtual work with external force F at the pipe wall interface and torque T at the spring-loaded joint, dh = nlcosθdθ

(7)

Fdh − Tdφ = 0

(8)

and

With the effective angular displacement of the torsion spring expressed as φ = π − 2θ

(9)

and based on the kinematic constraints shown earlier, the torque is T = kφ = k(π − 2θ)

(10)

with k being the spring constant, which gives F = 2k(π − 2θ)(nlcosθ)−1

(11)

This function has an interesting asymptotic behavior, as shown in Fig. 6, which represents a compromise between the “relaxing” spring and the “stiffening” kinematics as the leg extends and approaches kinematic singularity. The force (or apparent stiffness) can also be reduced from this maximum value by tensioning the leg retraction winch if desired.

Fig. 6. Maximum normal force generated at pipe wall (shown normalized with k/nl = 1 N, θ in radians, and F in N).

228

C. A. Nelson

A multibody dynamics model of the leg mechanism was built using ADAMS (MSC Software). As shown in Fig. 7, the model demonstrates the feasibility of the design concept in achieving functional leg deployment (i.e., it matches the kinematic model derived above). This wheel-leg can extend to nearly double the overall module length, and adding leg segments could further extend this reach. With dmin only slightly greater than w, this design satisfies the constraints outlined earlier in this section.

Fig. 7. ADAMS model of leg mechanism in compressed and extended positions.

5 Conclusions Reviewing the design requirements against the analysis and simulations given in the previous sections, it can be noted that: • The robot passively adjusts to small changes in pipe profile via the torsion spring in the wheel-leg • The robot actively adjusts to large changes in pipe profile by actuation of the legretraction winch • The robot accommodates large profile diameter changes through its multi-segment wheel-leg design which can extend up to several times the pipe size depending on the number of leg segments used • The robot accommodates gradual bends through the passive compliance of the wheelleg and sharp bends through active adaptation of wheel-leg extensions (see Fig. 3) The feasible design space was characterized through intersections of kinematic and geometric constraint inequalities, which can facilitate design for specific ranges of pipe diameter. It was also determined that this wheel-leg design possesses favorable properties in terms of force generation at the pipe wall interface. Validation of all of these findings using additional simulation and through prototyping and experiments is an important next step in this work. This should involve experimental verification of the wheel-leg kinematics and wheel-wall interaction forces, development of wheel control and locomotion/navigation strategies/algorithms, additional exploration of weight and

Preliminary Design and Modeling of a Robot

229

complexity tradeoffs related to the large number of independent degrees of freedom, and task-based testing of prototypes in simulated pipe environments.

References 1. Zhang, Y., Zhang, M., Sun, H., Jia, Q.: Design and motion analysis of a flexible squirm pipe robot. In: IEEE 2010 International Conference on Intelligent System Design and Engineering Application, pp. 527–531 (2010) 2. Venkateswaran, S., Chablat, D.: A new inspection robot for pipelines with bends and junctions. In: Proceedings of the IFToMM World Congress 2019, Mechanisms and Machine Science, vol. 73 (2019). https://doi.org/10.1007/978-3-030-20131-9_4 3. Fang, Y., Yang, F., Dong, Z.: Spatial transformation model and equations of detecting arms of wheel-type robot in cylindrical pipe. In: Proceedings of the IFToMM World Congress 2019, Mechanisms and Machine Science, vol. 73 (2019). https://doi.org/10.1007/978-3-030-201319_245 4. Kwon, Y.-S., Lim, H., Jung, E.-J., Yi, B.-J.: Design and motion planning of a twomoduled indoor pipeline inspection robot. In: IEEE International Conference on Robotics and Automation, Pasadena, CA, USA, May 2008, pp. 3998–4004 (2008) 5. Oya, T., Okada, T.: Development of a steerable, wheel-type, in-pipe robot and its path planning. Adv. Robot. 19(6), 635–650 (2005). https://doi.org/10.1163/1568553054255646 6. Zhang, Y., Yan, G.: In-pipe inspection robot with active pipe-diameter adaptability and automatic tractive force adjusting. Mech. Mach. Theory 42, 1618–1631 (2007) 7. Chang, W.-C., Huang, Y.-C., Chang, P.-Y.: Development of a 3D pipe robot for smart sensing and inspection using 3D printing technology. Smart Sci. 5(3), 123–131 (2017). https://doi. org/10.1080/23080477.2017.1338428 8. Hirose, S., Ohno, H., Mitsui, T., Suyama, K.: Design of in-pipe inspection vehicles for Ø25, Ø50, Ø150 pipes. In: IEEE International Conference on Robotics and Automation, Detroit, MI, USA, May 1999, pp. 2309–2314 (1999) 9. Roh, S.-G., Choi, H.R.: Differential-drive in-pipe robot for moving inside urban gas pipelines. IEEE Trans. Rob. 21(1), 1–17 (2005) 10. Lee, D., Park, J., Hyun, D., Yook, G.H., Yang, H.-S.: Novel mechanisms and simple locomotion strategies for an in-pipe robot that can inspect various pipe types. Mech. Mach. Theory 56, 52–68 (2012) 11. Nayak, A., Pradhan, S.K.: Design of a new in-pipe inspection robot. In: 12th Global Congress on Manufacturing and Management, GCMM 2014, Procedia Engineering, vol. 97, pp. 2081– 2091 (2014) 12. Ciszewski, M., Buratowski, T., Giergiel, M.: Track drives adjustment simulation for a versatile pipe inspection robot. In: Proceedings of the IFToMM World Congress 2019, Mechanisms and Machine Science, vol. 73. https://doi.org/10.1007/978-3-030-20131-9_197 (2019)

Sensitivity Analysis of Cable Actuations for Moving a Tensegrity Mechanism Along a Specified Path Pramod Kumar Malik, Keshab Patra, and Anirban Guha(B) Indian Institute of Technology Bombay, Powai 400076, India [email protected]

Abstract. Traditional robots are based on rigid links connected by prismatic and revolute joints. They suffer from high weight to load carrying capability ratios and inability to recover from damage. Tensegrity mechanisms, with their inherently high deformation tolerance, have the potential to overcome these difficulties. However, interconnectivity of many links through cables makes the control of movement of a link or node in a tensegrity more difficult than controlling a link of a traditional robot. Moving a node in a precisely controlled manner is necessary for locomotion. This is what has been attempted in this paper for walking gait generation. Four, six and eight bar symmetric tensegrity mechanisms have been investigated. The most sensitive cables elements of these structures which need to be actuated for moving ground touching nodes have been identified. The node movements were assumed to be slow enough to ignore vibration. Monte Carlo form finding method was used to get the stable positions at intermediate locations of a gait trajectory. It was found that only the four bar tensegrity mechanism allowed walking gait. All the other mechanisms having more than four bars show scraping at ground nodes which made gait generation practically infeasible. It was surmised that increased level of connectivity between nodes was responsible for this and an investigation of simultaneous cable and strut actuation was suggested for further investigation. Keywords: Tensegrity robots · Sensitivity analysis · Cable actuations

1 Introduction Robots have been traditionally constructed with rigid links. Electrical and hydraulic actuators are the traditional methods of actuation [1]. Designers of such robots have traditionally struggled with the problem of reduction of mass of links while maintaining structural integrity. Design of soft robots has been attempted to address this issue. Tensegrity mechanisms have the potential to address these drawbacks. Tensegrity mechanisms consist of tensile and compressive members [2]. The tensile members are called strings or cables and the compressive members are called struts or bars. Change of shape of these mechanisms can be achieved by changing the length of struts or changing the © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 230–237, 2021. https://doi.org/10.1007/978-3-030-58380-4_28

Sensitivity Analysis of Cable Actuations

231

length of cables. Change of length of a strut happens when it is replaced by a linear actuator while change of length of a cable can be achieved by reeling in or out a cable through a motor mounted at the end of a strut. Change of length in cables results in change in positions of nodes which can be used to generate a desired motion. It is important to identify a sequence of cable actuation which will create nodal displacement such that the tensegrity mechanism undergoes walking locomotion. Tensegrity based locomotive robot was first introduced by Kanchanasaratool and Williamson [3, 4]. They provided a solution to the path tracking problem through a passive nonlinear particle dynamics model. Jager and Skelton [5] have developed a non-linear model and linearized the nonlinear dynamics for a 2D tensegrity mechanism. But the method fails for 3D tensegrity mechanisms because the nonlinearity cannot be handled by their solution method. Aldrich et al. [6] have developed a dynamic model by considering the struts as fully rigid. However, this failed to reduce the difficulty level significantly due to nonlinearity in dynamics of the mechanism. To remove the above-mentioned difficulties, a user defined path has been created in this work and divided into small segments. Thereafter, a sequence of inputs (in terms of cable actuations) has been found out so that the tensegrity mechanism travels through the end points of each small segment step by step through quasi-static processes. This ensures that the tensegrity mechanism will be in equilibrium at the end of each small segment in following the path.

2 Gait Generation Technique in Tensegrity Mechanisms Even though walking is inefficient as compared to rolling, but it has its own advantage. If the robot needs to carry some object, then in most cases the object needs to be kept upright. This is very difficult with a rolling motion of a tensegrity based robot. A tensegrity based robot which can walk will not face this problem. Thus, the focus of this work is to make a tensegrity based robot walk along a specified path. In order to make a tensegrity mechanism walk, it is necessary to keep at least three nodes immobile and in contact with ground while allowing other(s) to move. The nodal displacements can be generated by actuating the cables. This paper searches for a sequence of cable actuation which will generate motion of a node along a predefined path. The actuation process has been considered quasi-static to avoid the problems occurring due to nonlinear dynamics. The path has been discretized into some finite segments as shown in Fig. 1 and the strategy for gait generation is very simple. Let the chosen path be denoted by ‘g’ and it is divided into ‘R’ small segments with endpoints (g1 , g2 , g3 , . . . , gR ). At each point ‘gi ’, the tensegrity mechanism is supposed to be in equilibrium and stable. Let the stable configuration at each state be denoted by ‘S i ’. We are interested in finding the sets of constant inputs (e1 , e2 , . . . eR ) in such a way that when the mechanism moves from ‘gi−1 ’ to ‘gi ’and for each ‘ei ’at ‘gi ’, the tensegrity will be still in equilibrium and stable. We have generated tensegrities using Monte Carlo based method described by Li et al. [7] for this study. A four bar mechanism, shown in Fig. 2 (a), has been taken as an example to show the sequence of cable actuation. The lengths of the bars are equal to 15 units and unstretched length of vertical and horizontal cables are 10 and 5 units respectively for the form-finding process. To avoid unnecessary cable actuation, sensitivity analysis

232

P. K. Malik et al.

has been performed to identify the cables which are more sensitive towards a particular ground node displacement.

Fig. 1. Discretised path followed by tensegrity mechanism

3 Pattern of Locomotion of Tensegrity Based Robots The general concept of movement of the four bar tensegrity mechanism has been borrowed from the walking pattern of a four-legged land animal.

Fig. 2. (a). Four bar tensegrity mechanism and (b) Stepwise locomotion of the base nodes of a four strut tensegrity mechanism

The four legged animals keep three legs in contact with the ground while moving the fourth leg. This pattern gets repeated to cover the whole path. The same pattern is also followed for the tensegrity based robot to cover the path. As mentioned above, the whole path has been discretized into small segments to reduce the nonlinear effect. Four-legged animals have four discrete steps in one complete gait cycle and the stepwise movement of the base of the tensegrity mechanism in X-Y plane has been shown in Fig. 2 (b). Four lower nodes (5, 6, 7, and 8) of the tensegrity mechanism are like the four legs of animals. In step 1, node 6 is moved forward keeping other nodes in ground, in step 2 node 8 is moved forward keeping other nodes in ground, in step 3 node 5 is moved forward keeping other nodes in ground and the same has been followed for node 7.

4 Sensitivity Analysis of Tensegrity Structure Sensitivity analysis was conducted to have an estimation of how the actuation(s) of a particular cable(s) affects the displacement of a particular ground node keeping all other ground nodes’ positions unchanged. The importance of sensitivity analysis is to identify the cables which generate a significant amount of displacement at every node. During actuation, the cables are actuated up to 1.5% of unstretched length for both increase and

Sensitivity Analysis of Cable Actuations

233

decrease in step sizes of 0.1% of the unstretched length. Walking in the forward direction without turning requires the trajectory of the ground nodes to be only in the XZ plane. Figure 3 shows the net magnitude of displacement of node 6 for actuation of the different cables. It is seen that cable numbers 5, 6, 7, 8 and 10 are the most sensitive ones for movement of node 6 in the XZ plane. During the upward and downward movements of node six, it is observed that there are no actuations in cables 11 and 12. These cables are not sensitive for the movement of node six. Hence, they not considered here. Again, the motions of node six in other planes like XY and YZ are not important for the kind of motion we want to generate. Therefore, the motions in other planes have not been considered here.

Fig. 3. Actuation of node 6 in X-Z direction

5 Simulation of Gait of Four Bar Tensegrities Each step of the gait cycle is divided into two sub-steps. In the first sub-step a particular node is moved forward with a small upward movement (along Z-direction) and in the second sub-step it is moved forward with a small downward movement so that the node will come to rest on the ground. A positive actuation value indicates increase in length and negative means decrease in length. In the first sub-step for node 6, it will be moved forward (along X-direction) with some upward movement (along Z-direction) after giving the specified cable actuations. In the second sub-step for node 6, it will be moved further in forward direction and will come to downward movement to rest on the ground, after giving the specified cable actuations as mentioned in Table 1. The net translating motion in X-direction after this step is 0.0406, whose top view is shown in Fig. 6. In similar way forward movements of other nodes with both upward and downward direction is represented for each sub-step in Table 1 respectively. After giving specified cable actuations, the net translation for nodes 8, 5 and 7 are found to be 0.0406, 0.0407 and 0.0417 respectively in X-direction. The nodal displacements of all the ground nodes in all three axes are shown in Fig. 4.

234

P. K. Malik et al. Table 1. Upward and downward movements of all the ground nodes

Cable Actuation of node 6

Actuation of node 8

Actuation of node 5 Actuation of node 7

No.

Up

Up

Up

Down

Down

Down Up

Down

1

0

0

0

0

0

0

0

0

2

0

0

0

0

0

0

0

0

3

0

0

0

0

0

0

0

0

4

0

0

0

0

0

0

0

0

5

0

−0.03

0

−0.05

−0.03

0.06

0.01

0

6

−0.05

−0.02

0

0

0.01

0.03

−0.01

0

7

0.05

0

−0.05

0

0

0

−0.04

0.04

8

0.01

−0.01

9

0

0

0

10

−0.04

0

−0.01

11

0

0

−0.02

12

0

0

0.04

−0.01 −0.02

0

0

0.01

0.02

0

0

0

0

0.005

0.005

0.005

−0.045

0.01

0.01

−0.005

0 −0.02

−0.02

0

0

0 0 0 0.015

Fig. 4. Movements of individual ground node in X, Y and Z directions

From the above result, it is clear that the movement of all the nodes in X-direction is same (within 2.5% range). At the end of the fourth step, the mechanism regains its initial configuration but in a different position which is ahead of the initial position. Thus, we can conclude that a gait cycle has been completed successfully and the mechanism has translated to a new position. These steps can be repeated to make the mechanism follow a predefined path as shown in Fig. 1. Figure 5 show the plots of X, Y and Z movements of each ground node vs normalized gait cycle time. The figures clearly show that the movement of each node is very close to the ideal movement. The small deviations seen in Y-motion of node 5 and X-motion of node 7 do not create a major change in trajectory of

Sensitivity Analysis of Cable Actuations

235

the gait. Figure 6 shows the movement of node 6 in the first step. Similar movements will occur to other ground nodes like 5, 8 and 7 in subsequent steps. Finally a new positions of all the nodes will be achieved.

Fig. 5. X, Y and Z movements of nodes in gait cycle

Fig. 6. Movements of node 6 in step 1

236

P. K. Malik et al.

6 Ground Node’s Movement for Six and Eight Noded Tensegrity Mechanisms The base of a six bar tensegrity whose all six nodes are in ground forms a hexagon shown in Fig. 7. Here the locomotion has been divided into 2 discrete steps. The red coloured nodes form one group and the black coloured nodes form another group. As each group consists of three nodes, in the first step the red coloured nodes will move forward with a very small step and the black nodes will be on the ground. When the red nodes will touch the ground, the black nodes will move forward and will regain its initial state as shown in Fig. 7. This process will repeat for covering the desired path. By these 2 steps the hexapod will complete one gait cycle.

Fig. 7. Six bar tensegrity mechanism and stepwise movements of base nodes.

A trial to move a single ground node was performed. The trial node is moved in the X direction by 0.002 units and in the Z-direction by 0.0505 units. The other nodes should not move but they also moved and the maximum movement among the movement of other nodes is 0.0105 units in X direction. We were unable to find a feasible combination of cable actuation which would move only the selected node without moving the other nodes. As the others nodes’ movement is undesirable, it was not possible to use it as a step in gait cycle. Simultaneous movement of two ground nodes in X-direction was also tried but was not possible without significant movement in the other nodes. When three nodes were moved in this trial, all three moved in the Z-direction by 0.0202 units but there was a relative displacement of 0.0116 units between them which is undesirable. So the gait generation by three node displacement was not possible. The base of the eight bar tensegrity whose all eight nodes are in ground forms an octagon as shown in Fig. 8. Here the locomotion can be divided into two discrete steps. The red coloured nodes form one group and the black coloured nodes form another group. These two steps can complete one gait cycle. An attempt was made to move a single ground node. A trial node was moved in the X-direction by 0.0158 units and in the Zdirection by 0.02 units. Unfortunately, this was accompanied by undesirable movement in the other nodes of which the maximum was a negative X-direction movement of 0.006 units. As movement of the other nodes are undesirable, gait generation through single node movement was not possible. According to the proposed method for the eight bar tensegrity mechanism, four ground nodes may move simultaneously to complete one step. Unfortunately, this too led to an undesirable 0.0173 unit movement between

Sensitivity Analysis of Cable Actuations

237

them. Thus, we are forced to conclude that it is probably possible to generate walking gait in a tensegrity based robot with only the four strut configuration. Any other more complicated tensegrity mechanism is likely to pose significant challenges to the control algorithm trying to generate a walk without ground scraping.

Fig. 8. Eight bar tensegrity mechanism and it’s base nodes.

7 Conclusion This paper explores the feasibility of walking gait generation in simple tensegrity mechanisms by cable actuations where the movement of nodes have been discretized into small steps. Movement at every such step was assumed to be slow enough to avoid problems associated with vibration. Our investigation was able to identify a feasible sequence of cable actuations for only a four bar tensegrity. For all the other tensegrity mechanisms with 6 and 8 bars, an attempt to move one or more ground nodes through a gait cycle led to significant unwanted movement in the other ground nodes. This would lead to ground scraping in real life. Thus, one is forced to conclude that high connectivity between nodes in complex tensegrity mechanisms makes walking gait difficult to achieve. However, a study involving simultaneous actuation of struts and cables and inclusion of friction at ground contact point is necessary to prove or disprove this hypothesis and such a study will be taken up in future.

References 1. Kasac, J., Novakovic, B., Majetic, D., Brezak, D.: Global positioning of robot manipulators with mixed revolute and prismatic joints. IEEE Trans. Autom. Control 51(6), 1035–1040 (2006) 2. Zhang, J., Ohsaki, M.: Tensegrity Mechanisms. Springer, Cham (2015) 3. Kanchanasaratool, N., Williamson, D.: Modelling and control of class tensegrity mechanisms. Int. J. Control 75(2), 123–139 (2002) 4. Kanchanasaratool, N., Williamson, D.: Motion control of a tensegrity platform. Commun. Inf. Syst. 2(3), 299–324 (2002) 5. De Jager, B., Skelton, R.E.: Input-output selection for planar tensegrity models. IEEE Trans. Control Syst. Technol. 13(5), 778–785 (2005) 6. Aldrich, J.B., Skelton, R.E., Kreutz-Delgado, K.: Control synthesis for a class of light and agile robotic tensegrity mechanisms. In: American Control Conference, 2003. Proceedings of the 2003, vol. 6, pp. 5245–5251. IEEE, June 2003 7. Li, Y., Feng, X.Q., Cao, Y.P.: Monte carlo form-finding method for tensegrity mechanisms. In: AIP Conference Proceedings, vol. 1233, no. 1, pp. 1112–1116. AIP, May 2010

An Approach to Motion Task-Oriented, Computer-Aided Design of Origami-Inspired Mechanisms Judith U. Merz1(B) , Felix J. Reimer2 , Mathias Hüsing1 , and Burkhard Corves1 1

Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University, Aachen, Germany {merz,huesing,corves}@igmr.rwth-aachen.de 2 fka GmbH, Aachen, Germany [email protected]

Abstract. The still young field of technical origami products offers considerable potential to give an innovative boost to stagnating product optimizations. The possibility to produce foldable patterns on planar, multifunctional material in two dimensions (2D), and thereof create – even by self-folding – three-dimensional (3D) structures enables a cost-effective and fast manufacturing. However, the complexity of designing transformable origami-based foldings proves to be challenging: there is a methodical gap between the use of known foldpatterns and their expected spatial motion, and the combination of such foldpatterns to generate a desired trajectory – e.g. for sealing, opening or moving. This leaves the creation of foldings to experts of kinematic modeling, who improve the system iteratively based on experience. Against this background, the following contribution presents a design tool for a computer-aided creation of foldings to fulfill given motion tasks. Following a target motion task, an optimization algorithm proposes solutions of origami foldings. The analysis is based on a numerical simulation of the folding motion. Keywords: Origami · Synthesis and development · Mechanism and robot design

1 Introduction In the context of innovative product development, the interest in integrating origami principles as an attractive alternative into the general design process is increasing. Thereby advantages of origami such as a compactability and reconfigurability, a high stiffness of thin facets, which are mutually supportive, and a high strength-to-weight ratio [1] are of great use. Especially the possibility to produce foldable patterns on planar, multifunctional material in 2D, and thereof create – even by self-folding – 3D structures, establishes a cost-effective and fast manufacturing [2–4]. Applications are in the design of movable origami robots [5] and transformable folded structures [6, 7] as well as for special materials with multistability [8], reprogrammability [9] or even as compliant mechanism joints [10]. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 238–246, 2021. https://doi.org/10.1007/978-3-030-58380-4_29

Approach to Motion Task-Oriented, Computer-Aided Design of Mechanisms

239

Origami-inspired rigid foldings, where stiff facets are folded along hinged creases, are capable of forming complex designs in various shapes and sizes. Yet, designing folding mechanisms is challenging because of complex kinematic dependencies and possible collisions during motion. For the technical realization of foldable mechanisms or deployable structures, extensive constructive, kinematic and kinetostatic analysis is required. This results in unusual challenges for general designer in the field of engineering and science. In order to establish origami-based structures as an alternative to conventional design concepts, design methods have to be easy to access and simplified by support of computer design tools. In this contribution, we propose a new approach for designing folding mechanisms or robots that move according to target trajectories. First, we present our methodology and a definition of input motion tasks (Sect. 3). Then we facilitate the design of origamibased folding mechanisms by introducing a computer-based optimization algorithm to calculate a folding for a specific motion task (Sect. 4). Finally, we demonstrate our results in a parameterized numerical simulation example (Sect. 5).

2 Related Work Design guidelines and frameworks, as used in various engineering fields, assist scientists and engineers to identify innovative but feasible solutions. First guideline proposals are presented for a general design synthesis process for folding mechanism [11] and a design methodology for robots and machines [12]. However, both procedures require particular skills synthesizing functionality and geometry in an iterative step. As an example, Felton et al. [4] describes how a folding for a moving robot was designed in a combination of kinematic simulations and iterative prototypes. Alternatively, known foldpatterns can be evaluated for their applicability to a given task. A collection and overview e.g. of the expected spatial motion and the combination of foldlines and foldfacets to form a foldvertex and ultimately to create a folding mechanism, is offered by Paris et al. [13]. A more general approach presents all foldable combinations to form pattern with the same facets by varying their facet geometric parameters, such as lengths and angles [14]. Vertices with either triangular or quadrilateral facets are considered. Comparably, Dieleman et al. [15] also varies lengths and partly angles for facets with four intersecting foldlines. That way puzzle stones are created which no longer share the same foldpattern with identical facets, but which are composed of different sized facets. In addition to this generalization, which has also been found by Tachi et al. [16], Dieleman et al. [15] defines a systematic approach via the puzzle stones to design foldable systems, which are at least able to reach two independently defined states. Further, also He et al. [17] describes the need for a computeraided, optimization-based design tool for the determination of different folding poses. Fields of research dealing with similar tasks are the theories of mechanisms and parallel robots. In this context, foldings are a special case, because firstly all foldvertices can be considered spherical mechanisms and secondly closed, kinematic chains. The foldlines can be regarded as ideal revolute joints (R). Abdul-Sater et al. [18] presents a so-called planar four-position and spatial three-position synthesis for a spherically constrained RR-chain. The motion task is based on three or four positions of a coordinate

240

J. U. Merz et al.

system defined as end effector with respect to the base coordinate system. During the synthesis, lengths and angles are analytically calculated for the system to be able to reach the given positions. The spherical coupling of the RR-chain is then constructed around the previously calculated conditions. The spherically coupled RR-chain can be seen as equivalent to two connected foldvertices.

3 Methodology Building on and supplementing design guidelines, as presented in Sect. 2, this contribution presents a computer-aided tool to develop a folding mechanism from a specified motion task. The synthesis design step requires a definition of the motion task as input and outputs a concept model (Fig. 1). A method to define the motion task is introduced in the following.

structural synthesis dimensional motion task optimization

concept model

constraints boundary conditions cost function

Fig. 1. For the design of a concept folding model a target motion task is required as input. Two design methods are presented.

point toleranced point orientation

z

toleranced orientation arbitrary connection trajectory x

y

off-limit region

Fig. 2. Modelling of the motion task of a guiding with given start and end points, a point with given orientation and a toleranced point as the beginning of a given trajectory to be fulfilled. The space dashed in the lower part is an off-limit region that should not be passed.

Approach to Motion Task-Oriented, Computer-Aided Design of Mechanisms

241

While Paris et al. [11] and Zhakypov et al. [12] perform the structural and dimensional synthesis in two separate, partly parallelized steps, the presented optimization algorithm combines structural and dimensional synthesis into one step. In addition, the motion task boundary conditions, constraints and the cost function need to be defined. At first a model is generated, subsequently the motion path of the model is evaluated and the cost function is computed. Finally, a best-fit model configuration is determined. The fold-specific design process is based on a definition of a target motion task. Figure 2 shows an example trajectory for a guiding motion. All given references for the motion task are described in the base coordinate system. The guiding input can be defined either as point (coordinates) or as orientation of the facet (pose). Besides, selecting start and end point or a desired path, regions to be passed through or to be avoided can be declared. Optionally the definition includes the facets orientation. Weighting of points or parts of paths offers the possibility to distinguish between demands and wishes defined in the requirement analysis. Further, if only single poses or points are required, the connection of these poses is considered arbitrarily.

4 Optimization Outline Based on the methodology an optimization process (Fig. 3) has been implemented in MathWorks’ MATLAB. The genetic optimization algorithm ga is used in this contribution, which minimizes a nonsmooth function with linear (in)equality constraints. In k iterations or – following the ga algorithm – over k successive generations the optimization function defines the best-fit configuration vector (see Subsect. 4.1). Thereby each generation g consists of a number of configuration vectors regarded to as population. START

Model setup Define number of vertices/parameters Define boundary conditions and constraints Define cost function for motion task

GA Algorithm

k iterations

Generate population of configuration vectors Call objective function for each configuration vector Generate model for current configuration Run MBS Analyze trajectory Evaluate cost function for each configuration vector check stop condition yes Return best-fit configuration vector no

END

Fig. 3. Algorithm design for the generation of a best-fit solution to a given motion task.

242

J. U. Merz et al.

Considering boundary and constraint conditions (see Subsect. 4.3) a multi-body simulation (MBS) model is automatically generated based on the current configuration vector v using the MATLAB-integrated folding mechanism library [13]. It provides tools to generate and run folding MBS models for given geometries and actuation configurations. The MBS outputs the trajectory related to the configuration vector v. The desired and current trajectory are then compared and its costs (see Subsect. 4.2) are calculated. Based on the analysis of previous generation’s costs the ga algorithm derives a new generation of configuration vectors. If a set number of generations is reached or an optimum has been found, the optimization terminates. Otherwise a new configuration vector is analyzed, which is derived from previous generations using ga’s genetic approach. 4.1

Optimization Variables

To ensure foldability the synthezised vertices in this contribution are based on the systematic design strategy of the jigsaw puzzle stones presented by Dieleman et al. [15]. Thus, only four-foldline vertices are considered, which are combined in a Kokotsaki mesh. In this contribution, the folding is limited to one puzzle stone configuration with a size of 3 × 3-facets (Fig. 4). By applying the puzzle stone strategy, we reduce the number of variables for one vertex to seven, as well as for a 3 × 3-folding to nine variables. The number of vertices is initially kept constant. Henceforth, n × m patterns are to be considered with a variable number of vertices.

t1 s1

t2

t3 s1

s2

s3

s2

s3

base end effector actuator

Fig. 4. 3 × 3-folding based on a puzzle stone configuration [15]; definition of lengths s1..3 , widths t1..3 and sector angles α1..3 , as well as base facet and end effector point.

To describe the trajectory of a folding, an end effector position or orientation vector has to be determined in reference to the base coordinate system. For the 3 × 3-folding the top left facet acts as fixed base, while the lower right facet holds the end effector in the most distant corner to the base (Fig. 4). In this contribution, the highlighted foldline is actuated along the angle δ . Based on the desired size of the folding mechanism and

Approach to Motion Task-Oriented, Computer-Aided Design of Mechanisms

243

the number of vertices, the configuration vector v is defined. The set of variables in v defines the current configuration. The vector v consists of the three sector angles α , the n lengths s and the m widths t. v = [ α1 ..α3 s1 ..sn t1 ..tm ]

(1)

4.2 Cost Function As a cost function, the mean deviation of the desired trajectory and the trajectory of the current test configuration is calculated. This has been implemented using the mean and maximum Euclidean distance. In addition, the cost function also includes the body volume, the assembly space of the folding mechanism, the maximum number of vertices/facets and the degree of freedom (DOF). Following the genetic approach the ga algorithm as described by Conn et al. [20] minimizes the sum of the products of weight w and cost term c, which is calculated for each generation g. In this approach only the mean and maximum deviation are evaluated in the cost function. n

min ∑ ci (g) · wi g

(2)

i=1

4.3 Boundary Conditions and Constraints Boundaries are defined for variables like the actuated folding angle δ , sector angle α lengths s and widths t as followed (Eq. 3–5). By limiting the angles the bistable flat position of the folding is avoided.

δ ∈ (0, π ) α ∈ (0, π ) si ,ti ∈ [0.1, 10]

(3) (4) (5)

The constraints define all quadrilaterals to be convex, so that concave and crossed quadrilaterals are neglected. For the sector angles the following boundary condition has been applied (Eq. 6). Unspecified crosses [19] are avoided by implementing Eq. 7. Further, special symmetry cases according to Dieleman et al. [15] are restricted (Eq. 8).

∑ αi = 2π

α1 + α2 = π ∧ α2 + α4 = π α1 + α2 < α3 + α4 ∧ α1 + α4 < α2 + α3 ∧ α2 > α4

(6) (7) (8)

5 Computation Example The optimization process (see Sect. 3) has been carried out for a pre-defined motion task (see Fig. 5, right, blue path). A number of test configurations have been included into the 3D trajectory presentation (see Fig. 5, right, grey paths). The final optimization result resembles the course of the motion input (see Fig. 5, right, red path). The resulting

244

J. U. Merz et al.

folding configuration is additionally presented (Fig. 5, left, folding states I–VI). The resulting folding is a valid combination of foldfacets and does not correspond to any known fold pattern: it is movable with one DOF, while lengths and angles of the facets are within a producible range as specified. Thus, the example in principle proves the functionality of the presented optimization process.

I

II

III

IV

V

VI

z

y x

Fig. 5. Optimization result example, left: best-fit solution in fold states I-VI; right: optimization result trajectory (red) for a pre-defined motion task (blue) and about 30 representative tested configurations (grey). Thereby the population size is 1000, the elite rate is 0.05, the crossover rate is 0.76 and the mutation rate is 0.19.

6 Conclusion and Future Work This contribution presents an approach for a computer-aided design tool of folding mechanisms based on a target motion task. Thereby the objective is to simplify the complex development of foldings in a way that they become accessible as an attractive alternative for general designs. Not only known foldpatterns are considered, but also their combination and their generalized variations. To this end, the used methodology was integrated into the design process and the motion task input was characterized. An optimization process has been implemented, which determines – based on the motion task and other boundary conditions and constraints – the dimensions of a folding. The calculation of the folding including the motion is done numerically in form of a multibody simulation in MATLAB SIMULINK. The application example proves the functionality of the optimization algorithm to determine foldings, which are able to follow a target path. Future work is dedicated to extend the scope of performance of the optimization tool. One intension is to include variable numbers of vertices or facets into the optimization algorithm. Additionally, end effector points or orientations should become an

Approach to Motion Task-Oriented, Computer-Aided Design of Mechanisms

245

output variable in accordance with the motion task. This may lead to smaller or even asymmetric foldings, because redundant connections can be eliminated. Further, the folding motion simulation is to be extended by a collision detection. So far, the motion is limited by the kinematic restrictions, which are caused by closed kinematic chains. Besides, not only one DOF, but also more DOF systems are to be computed in future. This requires a strategy for methodically distributing actuators. Other intensions are to improve quality and efficiency of the optimization algorithm e.g. extending the cost function.

References 1. Schenk, M., Guest, S.D.: Origami folding: a structural engineering approach. Origami 5, 291–304 (2011) 2. An, B., Rus, D.: Designing and programming self-folding sheets. Robot. Auton. Syst. 62(7), 976–1001 (2014) 3. Tolley, M.T., Felton, S.M., Miyashita, S., An, B., Rus, D., Wood, R.J.: Self-folding origami: shape memory composites activated by uniform heating. Smart Mater. Struct. 23, 9 (2014) 4. Felton, S., Tolley, M., Demaine, E., Rus, D., Wood, R.: A method for building self-folding machines. Science 345(6197), 644–646 (2014) 5. Firouzeh, A., Paik, J.: Robogami: A fully integrated low-profile robotic origami. J. Mech. Robot. 7(2), 021009 (2015) 6. Thün, G., Velikov, K., Ripley, C., Sauvé, L., McGee, W.: Soundspheres: resonant chamber. Leonardo 45(4), 348–357 (2012) 7. Miura, K.: Method of packaging and deployment of large membranes in space. Inst. Space Astronaut. Sci. Rep. 618, 1–9 (1985) 8. Silverberg, J.L., Evans, A.A., McLeod, L., Hayward, R.C.: Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014) 9. Waitukaitis, S., Menaut, R., Chen, B.G., van Hecke, M.: Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015) 10. Nelson, T., Avila, A., Howell, L., Herder, J., Farhadi Machekposhti, D.: Origami-inspired sacrificial joints for folding compliant mechanisms. Mech. Mach. Theory 140, 194–210 (2019) 11. Paris1, J., et al.: Kinematic and kinetostatic classification for motion-task-oriented synthesis of folding mechanisms. In: The International Meeting on Origami Science, Mathematics, and Education, Oxford (2018) 12. Zhakypov, Z., Paik, J.: Design methodology for constructing multimaterial origami robots and machines. IEEE Trans. Robot. 34(1), 151–165 (2018) 13. Paris2, J., et al.: Classification, collection, and simulation of folding mechanisms. In: 2018 International Conference on Reconfigurable Mechanisms and Robots (ReMAR), S. 1-8., Delft (2018) 14. Siebrecht, J., Corves, B., Paris, J., Dehn, S., Buffart, H., Jacobs, G., Weigel, C., Hoffmann, S., Trautz, M.: On using tessellation properties for the development of classifying criteria for foldable mechanisms. In: The International Meeting on Origami Science, Mathematics, and Education, Oxford (2018) 15. Dieleman, P., Vasmel, N., Waitukaitis, S., Hecke, M.: Jigsaw puzzle design of pluripotent origami. Nat. Phys. 16(1), 63–68 (2020) 16. Tachi, T.: Generalization of rigid foldable quadrilateral mesh origami. In: Proceedings of the International Association for Shell and Spatial Structures, Valencia (2009)

246

J. U. Merz et al.

17. He, Z., Guest, S.: Approximating a Target Surface with 1-DOF Rigid Origami (2019) 18. Abdul-Sater, K.: Task-Based. Computer-Aided Kinematic Design of Spherically Constrained Kinematic Chains, Thesis (2016) 19. Abel, Z., Cantarella, J., Demaine, E., Eppstein, D., Hull, T., Ku, J., Lang, R., Tachi, T.: Rigid origami vertices: conditions and forcing sets. J. Comput. Geom. 7, 171–184 (2015) 20. Conn, A.R., Gould, N.I.M., Toint:, Ph.L.: A globally convergent augmented lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Num. Anal. 28(2), 545–572 (1991)

Asymmetric Spatial Beams with Symmetric Kinetostatic Behaviour Ali Amoozandeh Nobaveh(B) , Giuseppe Radaelli , and Just L. Herder Precision and Microsystems Engineering Department, Delft University of Technology, 2628CD Delft, The Netherlands [email protected]

Abstract. A method to achieve symmetric kinetostatic behaviour in an extensive working range at the endpoint of an asymmetric spatial beam, using cross-section optimization, is presented. The objective function of the optimization is defined as expanding the beam working range to the desired region, simultaneously maximizing symmetric behaviour in it. To reach this goal, a beam with predefined spatial global shape and an ‘I’ cross-section selected. The cross-sectional dimensions throughout the beam are used as input values for the optimization. The endpoint displacements under symmetric loadings are attained using a nonlinear co-rotational beam element based on the Euler-Bernoulli beam formulation. The optimized beams are compared to a circular cross-section beam with the same global shape to show the efficacy of the method. Isoforce diagrams are investigated for the optimized beams to show the symmetry behaviour of the beam endpoint and the effect of changing different parameters in cross-sectional optimization is discussed. Keywords: Spatial compliant mechanism · Asymmetric beam · Symmetric stiffness · Cross-section optimization · Isoforce mapping

1

Introduction

The ability to design spatial elastic elements with a predetermined kinetostatic character, i.e. the integration of the function of kinematic and spring elements, is desirable in several application domains. In wearable device technology, for example, it is imperative to design parts that are able to apply support forces to the body in concordance with specific kinematics of the limbs. It is presumably redundant to say that other application domains may partially profit from these functional requirements as well, ranging the entire spectrum of application of robotics and mechanisms. In recent years, spatial compliant mechanisms have increasingly been the topic of interest in the mechanism design community. Efforts have been made in both the characterization as in the design and optimization of spatial compliant mechanisms for the sake of advancing their applicability as full devices Supported by NWO (P16-05: Shell Skeletons). c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 247–254, 2021. https://doi.org/10.1007/978-3-030-58380-4_30

248

A. Amoozandeh Nobaveh et al.

or as components. In flexures, which are flexible elements widely used for precision mechanisms, there are plenty of examples of spatial mechanisms [6,10] and established design principles exist [4,12]. In these applications the kinematic requirement is leading, i.e. the contrast in stiffness between free and constraint directions must be extreme, on the other hand, ranges of motion are often small and the spatial occupation of the mechanism is not restrictive. This often leads to the employment of straight or notch flexures, in relatively complex spatial orientations and topologies with many parallelly connected kinematic chains. Examples of spatial compliant mechanisms with large ranges of motion and simpler topologies, e.g. [2,7–9,11,13,14], are emerging, but scarcer. From an utilization perspective as well as from a fabrication perspective, it appears convenient to consider an utmost simple topology of a cantilevered spatial beam, that is, a beam clamped at one end and loaded at its opposite end. The peculiar kinetostatic behaviour sought in this work is one where the spatial beam has a plane symmetry of loading forces and corresponding endpoint displacements, while the geometry of the beam itself is not symmetric with respect to the same plane. This type of behaviour has not been found by the authors in any earlier work. On the other hand, it is expected to be useful in distinct applications where the plane or area of action of the end-effector does not coincide with the available grounding positions. This is true for manipulatortype devices, where the ground attachment point must not interfere with the end-effector range of motion, and also in wearable devices where the symmetry plane of the body is to a large extend occupied by the body itself. The usual behaviour of whichever spatial beam is that upon loading the free end with a force, it somehow turns about the ground attachment point as a result of the moment produced by that force. It is hypothesised that there exist nonuniform distributions of cross-sectional properties for which the natural turning behaviour of the beam is corrected towards a more planar motion. This paper presents an asymmetric spatial cantilevered beam with a semisymmetrical kinetostatic behaviour, generated by optimization of cross-sectional properties of a given global beam shape. The cross-sectional shape is chosen to be a conventional I-profile. The symmetric behaviour is optimized by minimizing the differences in point-pairs displacements under symmetric loading about the desired plane. The beam displacements are computed numerically with a finite element model for beams, suitable for large displacements. The paper is structured as follows. In Sect. 2 the definition of the global geometry of the beam is given together with the description of the cross-sectional parameters to be optimized, the formulation of the objective function and the optimization settings. In Sect. 3 the results are shown and compared to the nonoptimized beam. A discussion on the validity and applicability of the results and possible improvements is given in Sect. 4, and the conclusion is given in Sect. 5.

Asymmetric Spatial Beams with Symmetric Kinetostatic Behaviour

2

249

Methods

The global undeformed beam shape (see Fig. 1a), could be chosen arbitrarily and will not be subject to optimization. In this particular case the reference axis of the beam is parametrized as 2

(x, y, z) = (C1 t3 , a − b(C2 t − c) , C3 t),

(1)

where t is the independent parameter ranging √ from 0 to 40, the constants a, b and c are chosen as a = 1.5, b = 0.75, c = 2, and the constants C1 , C2 and C3 are determined such that the end of the beam reaches the arbitrarily chosen coordinates Xe = 0.20 m, Ye = 0.15 m and Ze = 0.50 m. This is achieved by satisfying the relations  a−Ye +c Xe Ze b (2) , C3 = . C1 = 3 , C2 = t t t

dy +120 mm

Symm. plane

Symm. plane

Fy

Fx

Fy -α α

Fx dx +40 mm

-40 mm φ b

H

h B

(a)

(b)

Fig. 1. (a) The global shape of the beam and its projections on the main planes, (b) parameters of the ‘I’ cross-section, and symmetric loadings in desired working range of the beam.

The beam is clamped at the lower end and is loaded by force F at the endpoint. The magnitude of the force, is kept constant, and the components of the force vary with the angle α, defined from the plane of desired symmetry, as (fx , fy , fz ) = (F sin α, F cos α, 0).

(3)

250

A. Amoozandeh Nobaveh et al.

The defined objective function f for the optimization procedure includes two parts f1 and f2 . The first part, f1 , consists of the average of displacement differences in the three Cartesian coordinates for a number of point-pairs, resulting from a mirrored loading condition with a force magnitude of F = 100 N and an angle of ±α, where α = 30◦ , 60◦ , and 90◦ . This error part is named symmetry error. This part of the objective assures a degree of symmetry of the kinetostatic response within the region reached by the applied forces. However, that region might be unsatisfactory in terms of in-plane and out-of-plane displacement magnitudes, i.e., the resulting symmetry could be achieved only in minimal displacement ranges or in narrow bands of offset from the symmetry plane. For this reason, a second part of the objective, f2 , is defined. The second part considers the error of the end point displacement under the loadings in the x and y directions concerning chosen working range, in this case, 0.12 m in the y direction and 0.04 m in the x direction (see Fig. 1b). This error part is named range error. The total objective function is an unweighted summation of the above-mentioned parts and is defined as

f = f1 + f2 = (



1 (|dx+α − dx−α | + |dy+α − dy−α | + |dz+α − dz−α |)) 3 α=30,60,90 + (|0.04 − dxα=90 | + |0.12 − dyα=0 | + |dxα=0 |), (4)

where α is the effective angle of F and dx, dy, dz are displacements in the three coordinate directions. The optimization procedure is assigned to minimize the mentioned objective function by optimizing the cross-sectional properties of the described beam. In principle, it is possible to choose any cross-section for the spatially curved beam. However, the I-shaped cross-section is selected among prevalent commercial cross-sections, since changing its parameters enables a large variety of combinations of second moments of inertia in the two main axes, and the torsional constant. The cross-sections are doubly symmetric and are defined by their web height H, flange width B, flange thickness to web ˜ = h/H, web thickness to flange width ratio ˜b = b/B, and the height ratio h orientation φ (see Fig. 1b). The cross-section parameters are optimized at N main cross-sections (see Fig. 3) and interpolated by a B-spline to find out all other cross-section properties along the beam length. The employed interpolating B-spline is a degree-four for five or more cross-section and degree-two for three cross-sections, associated with an open uniform knot vector with its internal knots determined based on de Boor algorithm [5]. This knot vector ensures that the parameters of the first and last optimized cross-sections coincide with the first and last cross-sections of the beam itself. It is important to note that optimizing all cross-sections separately could lead to more promising results. However, it would cause more expensive computations, and notable dimension discontinuities between the beam elements which make the utilized mechanical model invalid.

Asymmetric Spatial Beams with Symmetric Kinetostatic Behaviour

251

The displacements mentioned in Eq. (4) are derived from a self-developed finite element solver using geometrically nonlinear co-rotational beam elements introduced by Battini [1], based on a Euler-Bernoulli beam formulation, and the described beam is discretized into forty elements. For the optimization process, the Multi Start option with five random starting points, using the fmincon function with the Interior-Point Algorithm from R optimization toolbox is used. The optimization was subject to the the Matlab bounds on the parameters shown in Table 1 as Min and Max. These bounds are set to limit the algorithm to come up with reasonable dimensions and to keep the I-shape through the beam. The material constants used in the model are Young’s modulus E = 200 GPa and shear modulus G = 76.9 GPa.

3

Results

To evaluate the method, beams with different optimized cross-sections but identical global shape are subjected to the same loading conditions as elaborated in Sect. 2, and the objectives are compared. To represent the kinetostatic behaviour of the beam, an isoforce mapping [3] is presented. Each isoforce line represents the displacement of the endpoint of the beam subject to a constant magnitude of the force F (see Fig. 1b), with a full cycle of the angle α. This mapping is shown for a reference beam, with circular cross-section and optimized radius of 5.4 mm, (see Fig. 2a and d). The red points on the x and y axis represent the positions of the desired displacements when the force is applied only in the x and y directions respectively, while the blue points represent the actual displacements obtained when the beam is subjected to these loadings. The green dashed lines connect the displacement of pairs of points, i.e., with opposite angle α of force F , within the optimized region. It also shows the cross-sections, in the undeformed and deformed configurations. This figure illustrates how straight the endpoint of the beam moves subject to a force in the y direction only (see Fig. 1a). It can be observed that with a circular cross-section the desired displacement is poorly achieved and that a force in the y direction results in a notable lateral drift. Similar figures are provided (see Fig. 2b, e and Fig. 2c, f) for an I-beam with a uniform optimized cross-section (N = 1), and another I-beam with five optimized main cross-sections (N = 5). The optimized parameters are given in Table 1. An analysis of the impact of the number of optimized cross-sections on the resulting objective was performed (see Fig. 3). Also, as a benchmark, the objective achieved by a uniform circular cross-section is shown to highlight the performance leap obtained by the optimization effort.

4

Discussion

The improvements on the symmetry behaviour obtained by optimizing crosssectional parameters across the I-beam are significant (see Figs. 2b and c) as compared to circular beam. The obtained displacements have a undoubtedly

252

A. Amoozandeh Nobaveh et al.

0.3

0.3

0.3

0.25

0.25

0.25

0.2

0.2

0.2

0.15

0.15

20 60 100

0.1

0.15 20

20

0.1

0.1 60

60

0.05

0.05

0.05 100

100

0

0

0.05

0.1

0.15

0.2

0.25

0

0

0.05

(a)

0.1

0.15

0.2

0.25

0

0.6

0.55

0.55

0.55

0.5

0.5

0.5

0.45

0.45

0.45

0.4

0.4

0.4

0.35

0.35

0.35

0.05

0.1

0.15

(d)

0.2

0.25

0.3

0

0.05

0.1

0.1

0.15

0.2

0.25

0.15

0.2

0.25

(c)

0.6

0

0.05

(b)

0.6

0.3

0

0.15

0.2

0.25

0.3

(e)

0

0.05

0.1

(f )

Fig. 2. Isoforce mapping in the desired range, (a, d) circular beam top and rear view (b, e) uniform ‘I’ cross-section beam top and rear view (c, f) non-uniform ‘I’ cross-section beam top and rear view. Cross-section dimensions are mentioned in Table 1.

0.14 I-beam total error I-beam range error I-beam symmetry error Circular beam total error Circular beam range error Circular beam symmetry error

0.12

Objective (m)

0.1 0.08 0.06 0.04 0.02 0

1

3 5 7 Number of optimized cross-sections

9

Fig. 3. Objective function for circular, uniform and non-uniform optimized ‘I’ beam.

Asymmetric Spatial Beams with Symmetric Kinetostatic Behaviour

253

Table 1. Optimized parameters for uniform cross-section and non-uniform crosssections optimization with bounds. Parameter Min Max H(mm)

4

B(mm) ˜ h ˜b φ(deg)

CS

CS-1 CS-2 CS-3 CS-4 CS-5

8

5.75

5.4

6.1

6.3

6.3

6.2

6

12

6.77

6.5

7.6

8.8

9.2

9.2

0.2

0.8

0.46

0.47

0.48

0.53

0.55

0.53

0.2

0.8

0.79

0.64

0.58

0.54

0.51

0.50

10

170 85.96 89.98 89.99 89.99 90.00 90.00

better match, the region of symmetry is more extended, and the path of the endpoint subject to a force in the y-direction is straighter. It is fair to note that the symmetry in z-direction was not improved with the optimization. Apparently, the effect of the global shape and its inclination is hard to counteract by the cross-section variations allowed by the given model. This also underlines the dependence of the presented results to the initially chosen global shape. The proposed procedure could generally be applied to every global geometry, but the results heavily rely on it. Depending on the flexibility of the design space of the specific problem at hand, it might be recommendable to employ a combined optimization of global shape and cross-sections. The optimization process shows that increasing the number of optimized cross-sections will not indeed lead to lower errors yet make the problem computationally expensive. This optimal number of optimized cross-sections should be investigated for each desired range and initial conditions. There are limitations inherent to the employed mechanical model which follow from the Euler-Bernoulli assumptions and from the required symmetry of the cross-sections, of which the effect still has to be investigated. Relieving these assumptions, which suppress all cross-sectional deformations, e.g., shear deformation, warping and in-plane deformation, presumably makes more complex behaviour emerge, possibly allowing an enhanced performance as a result.

5

Conclusion

This paper presents a new method to achieve symmetric kinetostatic behaviour from asymmetric spatial beams using cross-sectional optimization, given a global shape. The global shape could be designed with a specific path or line based on requirements. The effectiveness of this method has been validated by comparing symmetry error and range error of the optimized conventional ‘I’ beam and circular beam cross-section in a predetermined field. Rather complex crosssectional properties were found by optimization, using a nonlinear beam element to determine displacement differences between point pairs and minimizing them, which resulted in an symmetric behaviour in a proportionally extensive working range. Such design requirements are not easily achieved with existing methods for

254

A. Amoozandeh Nobaveh et al.

compliant mechanism design, which shows the capacity of this method to handle more complex design demands.

References 1. Battini, J.M.: Co-rotational beam elements. Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden (2002) 2. Bilancia, P., Berselli, G., Bruzzone, L., Fanghella, P.: A CAD/CAE integration framework for analyzing and designing spatial compliant mechanisms via pseudorigid-body methods. Robot. Comput. Integr. Manuf. 56, 287–302 (2019). https:// doi.org/10.1016/J.RCIM.2018.07.015 3. Doornenbal, B.: Zero stiffness composite shells using thermal prestress. MSc thesis, Delft University of Technology, Delft, The Netherlands (2018) 4. Hopkins, J.B., Culpepper, M.L.: Synthesis of multi-degree of freedom, parallel flexure system concepts via Freedom and Constraint Topology (FACT) – Part I: Principles. Precis. Eng. 34(2), 259–270 (2010). https://doi.org/10.1016/ J.PRECISIONENG.2009.06.008 5. Hoschek, J., Lasser, D.: Fundamentals of Computer-Aided Geometric Design. A.K Peters, Mass (1993) 6. Howell, L.L., Magleby, S.P., Olsen, B.M.: Handbook of Compliant Mechanisms, 1st edn. Wiley, Somerset (2013) 7. Nijssen, J.P.A., Radaelli, G., Herder, J.L., Ring, J.B., Kim, C.J.: Spatial concept synthesis of compliant mechanisms utilizing non-linear eigentwist characterization. In: Volume 5A: 42nd Mechanisms and Robotics Conference. American Society of Mechanical Engineers, August 2018. https://doi.org/10.1115/DETC2018-85307 8. Parlakta¸s, V.: Spatial compliant constant-force mechanism. Mech. Mach. Theory 67, 152–165 (2013). https://doi.org/10.1016/j.mechmachtheory.2013.04.007 9. Parlakta¸s, V., Tanık, E.: Single piece compliant spatial slider-crank mechanism. Mech. Mach. Theory 81, 1–10 (2014). https://doi.org/10.1016/J. MECHMACHTHEORY.2014.06.007 10. Rad, F.P., Berselli, G., Vertechy, R., Castelli, V.P.: Compliance based characterization of spherical flexure hinges for spatial compliant mechanisms. In: Advances on Theory and Practice of Robots and Manipulators, pp. 401–409. Springer (2014). https://doi.org/10.1007/978-3-319-07058-2 45 11. Radaelli, G., Herder, J.L.: Study on the large-displacement behaviour of a spiral spring with variations of cross-section, orthotropy and prestress. Mech. Sci. 9(2), 337–348 (2018). https://doi.org/10.5194/ms-9-337-2018 12. Turkkan, O.A., Venkiteswaran, V.K., Su, H.J.: Rapid conceptual design and analysis of spatial flexure mechanisms. Mech. Mach. Theory 121, 650–668 (2018). https://doi.org/10.1016/J.MECHMACHTHEORY.2017.11.025 13. Weeger, O., Narayanan, B., Dunn, M.L.: Isogeometric shape optimization of nonlinear, curved 3D beams and beam structures. Comput. Methods Appl. Mech. Eng. 345, 26–51 (2019). https://doi.org/10.1016/j.cma.2018.10.038 14. Zhou, H., Ting, K.L.: Geometric optimization of spatial compliant mechanisms using three-dimensional wide curves. J. Mech. Des. 131(5), 051002 (2009). https:// doi.org/10.1115/1.3086792

A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator Naoto Kimura(B) and Nobuyuki Iwatsuki Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan [email protected]

Abstract. A semi-automatic method for type synthesis of a closed-loop spatial path-generator with 1 degree of freedom (DOF) is proposed. By using this methodology, the suitable number of links and the suitable selection of kinematic pairs can be derived to generate the specified trajectory precisely. This methodology has several steps. Firstly, a serial kinematic chain is specified according to the desired task. Then, the number of links of another serial chain added to it to build a closedloop is determined based on the number of designated output points and mobility. Next, both suitable selection of kinematic pairs in the added chain and optimal dimensions of the linkage are derived at the same time based on the output tracking errors and the force transmission. In order to investigate the performance of this methodology, an example of a spatial-path generator is synthesized. Keywords: Spatial linkage · Type synthesis Automatic synthesis · Transmission index

1

· Optimal synthesis ·

Introduction

Path-generating linkages with 1 DOF are used for many machines, such as industrial machines, walking robots and so on, because of their simple structures and high reliability. In order to synthesize a path-generating linkage, number synthesis, structural synthesis and dimensional synthesis are generally performed in order. However, structural synthesis generally has to be performed with designers’ knowledge and experiences. In order to synthesize the structure of a pathgenerating linkage easily, several researchers have proposed automatic methods of structural synthesis [1–4]. These methods optimize both the structure and dimension of a linkage at the same time so as to generate the specified trajectory precisely. However, these methods are just for planar linkages, and methods for spatial path-generating linkages have not been proposed. Since a spatial linkage can have various types of kinematic pairs, synthesizing it is a difficult problem. Besides, the conventional methods can synthesize an impractical linkage with a complex structure and low force-transmission. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 255–263, 2021. https://doi.org/10.1007/978-3-030-58380-4_31

256

N. Kimura and N. Iwatsuki

Fig. 1. Coordinate systems on an example of a spatial-path generator

Fig. 2. Displacement between two links in a spatial linkage

Therefore, in this paper, a semi-automatic method to synthesize a spatial path-generator with various types of lower pairs is proposed. The linkage has a single closed-loop topology to simplify the structure. In this method, a serial kinematic chain in the closed-loop is specified according to the desired task, and another part is synthesized automatically. This is the concept of the “semiautomatic” type synthesis, and it can simplify the design while retaining the creativity of the designer. In the automatic process, a suitable selection of kinematic pairs and optimal dimensions are derived so that the linkage can generate the desired trajectory precisely with high force-transmission. In Sect. 2, a kinematic model of the linkage is proposed. In Sect. 3, the method of a semi-automatic type synthesis is proposed. In Sect. 4, the optimization problem is formulated with a transmission index. In Sect. 5, a design example is shown.

2

Generalized Kinematic Model

A kinematic model of a closed-loop spatial path-generator is proposed, where kinematic constraints of kinematic pairs are parameterized to represent all types of typical lower pairs. At first, dimensions of the linkage are parameterized. Figure 1 shows an example of the linkage. The global coordinate system is Σ0 . The local coordinate system fixed on the i-th pair is Σi . Dimensions of the linkage are parameterized with the initial posture, i.e. position pi = [xi yi zi ]T and direction φi = [0 φy,i φz,i ]T (Euler angles1 ) of the i-th pair when displacement of the pair is zero. Next, the displacement between two links at each pair is parameterized. When two links displace with each other at the i-th pair, Σi is separated into Σi,A and Σi,B as shown in Fig. 2. Then, the translation and rotation displacement are represented as di = [ai,1 dx,i 0 0]T and Ri (ai,2 θx,i , ai,3 θy,i , ai,4 θz,i ), respectively. Ri is the rotation matrix from Σi,A to Σi,B , and ai,j is a binary value with 0 or 1. Because of this parameterization, all types of typical lower 1

Two Euler angles are enough to specify the direction of a rotation/translation axis.

A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator

257

Fig. 3. Proposed procedure of a semi-automatic type synthesis

pairs can be represented with the value of ai = (ai,1 , ai,2 , ai,3 , ai,4 ). For example, ai = (1, 1, 0, 0) represents a cylindrical pair. By using the dimension parameters of the i-th and the (i + 1)-th pair, the homogeneous matrix Li,i+1 which represents the coordinate transformation from Σi,B to Σi+1,A can be derived. Besides, by using the displacement parameters of the i-th pair, the homogeneous matrix Pi which represents the coordinate transformation from Σi,A to Σi,B can be derived. Thus, the closed-loop equation of the linkage is represented as the following equation. N  i=1

Li−1,i Pi = L−1 N,0 ,

(1)

where N is the total number of pairs. When a local coordinate system Σr is fixed at the output point, the homogeneous matrix of the output point D0,r can be derived. Then, the output position pr on Σ0 can be obtained from D0,r .

3

Semi-automatic Type Synthesis

A semi-automatic method of type synthesis is proposed. Figure 3 shows the proposed design procedure. At first, the desired output trajectory is specified as several designated points. Then, a serial kinematic chain which can generate the specified trajectory is specified according to the desired task, and another serial chain is added to it so as to make a closed-loop. Here, the only number of links in the added chain is systematically determined. This process is called the adding chain process. Next, optimization conditions are specified, and both a suitable selection of kinematic pairs in the added chain and optimal dimensions of the linkage are automatically derived. This process is called the mobility restriction process. Finally, accuracy and force transmission of the linkage are evaluated.

258

N. Kimura and N. Iwatsuki

Fig. 4. The adding chain process and the mobility restriction process

3.1

Adding Chain Process

The conceptual diagram of the adding chain process is shown in Fig. 4, where a serial chain is added to make a closed-loop with the specified serial chain. The number of links and kinematic pairs in the added chain is calculated to satisfy the condition of mobility and the number of designated points. The relationship between mobility and number of loops in a general parallel mechanism is F = FJ − 6L, where F is the output mobility, FJ is the sum of mobility of kinematic pairs and L is the number of loops. Since the linkage has F = 1 and L = 1, FJ = 7 holds. When the sum of mobility of kinematic pairs in the specified serial chain is Fc , the number of kinematic pairs in the added chain Na has to satisfy the following condition. Na  7 − Fc

(2)

In order to have sufficient design parameters to generate the specified motion precisely, a relationship between the number of designated points and design parameters are considered. It is assumed that 6 DOF of the added chain are constrained when the output point of the linkage passes through a designated point. Since the output point passes through the Np designated points and the initial position, the total number of the constraints is 6(Np + 1). The total number of dimension and displacement parameters in the added chain are Nd Na and (7 − Fc )Np , respectively. Then, Nd Na + (7 − Fc )Np has to be larger than 6(Np + 1) to path through all designated points. Besides, subordination between displacement and design parameters also has to be considered. Posture angles of the i-th pair, φy,i and φz,i , can be subordinate to the displacement angles when a spherical pair is used. Since the maximum number of kinematic pairs with multiple DOF is [(7 − Fc ) − Na ], the maximum number of subordinate parameters is represented as 2(7 − Fc − Na ). Therefore, the following inequality holds. Nd Na + (7 − Fc )Np − 2(7 − Fc − N a)  6(Np + 1)

(3)

A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator

Therefore, Na is determined as follows to satisfy Eqs. (2) and (3).  ceil(ζa ) (ζa < 7 − Fc ) (Fc − 1)Np + 2(10 − Fc ) Na = ), (ζa = Nd + 2 7 − Fc (ζa  7 − Fc )

259

(4)

where ceil(x) = min{n ∈ N|n  x}. 3.2

Mobility-Restriction Process

In order to synthesize a superior mechanism, it is desired that kinematic pairs in the added chain are selected to minimize errors between the desired and the actual output trajectory. Thus, both calculating dimension and selecting kinematic pairs are performed at the same time. Since the design parameters have both continuous and binary values in this problem, the optimization is performed with the following procedure. (1) A list of lower pairs used for the linkage is specified as shown in Fig. 4. (2) Kinematic constraints of all pairs in the added chain shown are assumed as ai = (1, 1, 1, 1) (i = 1, 2, ..., Na ). This state is the best for the linkage to generate the desired motion because its mobility is the largest. (3) A kinematic pair is picked from the list and applied to the i-th pair. (4) Dimensional synthesis is performed to minimize an objective function. (5) The process from (3) to (4) is performed for the all kinematic pair in the list, and the kinematic pair which gives the minimum objective value is selected as the i-th kinematic pair. (6) The processes from (3) to (5) are applied for all kinematic pairs in the added chain in order. Note that the obtained selection of kinematic pairs is not optimal but suitable at least. In order to get a linkage with 1 DOF, kinematic pairs are chosen considering mobility in the above process (3). When a kinematic pair with Fi DOF is picked for the i-th pair, summation of mobilities of from (i + 1)-th to Na -th pair is (7 − Fc ) − (Fi + Fsum,i−1 ), where total mobility of from the first to the (i − 1)-th selected pair is Fsum,i−1 . Since mobility of typical lower pairs is from one to three, the following inequality holds. Na − i  (7 − Fc ) − (Fi + Fsum,i−1 )  3(Na − i)

(5)

If a kinematic pair satisfying Eq. (5) is chosen from the kinematic pair list in the process (3), the linkage finally can have 1 DOF.

4

Dimensional Synthesis

An objective function for dimensional synthesis in the mobility restriction process is formulated. In this optimization, the error between the desired and the

260

N. Kimura and N. Iwatsuki

actual output trajectory is minimized, and a transmission index is maximized at the same time to get a practical linkage. As a transmission index for a parallel mechanism, the output transmission index (OTI) has been proposed [5] and can be represented as shown below in the case of a single closed-loop linkage. OTI =

SrT ◦ SO , |SrT ◦ SO |max

(6)

where SrT is the wrench screw which is transmitted from the input to the output, and SO is the output twist screw. Although |SrT ◦ SO |max can be calculated with a geometrical relationship between the two screws [6], the calculation is a little complicated. Therefore, a different definition of OTI is proposed in this paper. In this definition, all screws are replaced by the following vector.  ˆ = √1 u S 2

p×u |p×u |

T

,

(7)

where u is a unit direction vector and p is a position vector of each pair and the ˆ = [u 0]T output point. Note that a pure rotation and a pure translation are S ˆ = [0 u]T , respectively. Then, Eq. (6) is rewritten as the following equation. and S ˆr ◦ S ˆO OTI = S T

(8)

When the output angular velocity and translation velocity have same direction with transmitted torque and force, respectively, the force transmission becomes maximum. In contrast, when they are orthogonal, the force transmission becomes minimum. The OTI of Eq. (8) becomes one in the maximum force transmission and zero in the minimum force transmission. Therefore, this index can evaluate the force transmission of the linkage. An objective function for the multi-objective optimization with output position errors and the OTI is constructed. In order to reduce the calculation cost, both displacement and dimension of the linkage are optimized at the same time. Let c is a vector including all displacement and dimension parameters. Then, the objective function is set as the following formula. Np 1  [w1 |pr,j,ref − pr,j |2 − w2 OTIj ], f (c) = Np j=1

(9)

where w1 and w2 are weight values to specify the trade-off between accuracy and force transmission, and pr,j,ref is the specified output position on the desired trajectory. This objective function is minimized with a numerical optimization method subjecting to the closed-loop equation of Eq. (1), the design space cmin  c  cmax , the minimum size of each link (xi − xi−1 )2 + (yi − yi−1 )2  lmin . The other conditions can be also added depending on the application.

A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator

5

261

Design Example

As an example, the synthesis where a suitable kinematic chain is added to the serial chain like human-arm shown in Fig. 5 so as to generate the specified pick and place motion with 1 DOF was performed. The specified chain has a revolute pair and a spherical pair. The specified initial position was p0 = [500 0 0]T . As designated points, the picking position pr,1 = [500 250 − 100]T and the placing position pr,2 = [500 250 − 100]T were specified. Firstly, adding chain process was performed. Substituting the mobility of the specified chain Fc = 4, the number of designated points Np = 2 and the number of design parameters Nd = 5 for Eq. (4), the number of pairs in added serial chain can be calculated as Na = 3. Therefore, a serial chain with two links and three pairs is added to the link2 shown in Fig. 5.

Fig. 5. The specified serial chain in an example of a linkage

Fig. 6. Output transmission index of the synthesized linkage

Fig. 7. The motion of the synthesized spatial path-generator

Next, the mobility restriction process was performed. The specified kinematic pair list had a revolute, prismatic, cylindrical, and spherical pair. Weight values

262

N. Kimura and N. Iwatsuki

of the objective function were w1 = w2 = 1. The minimum size of links was lmin = 30 mm. Design space of the dimension was [−300, 0, −500, −2π, −2π]  [xi , yi , zi , φy,i , φz,i ]  [300, 0, 500, 2π, 2π], where unit of position and angle was mm and rad, respectively. In order to drive the linkage with a motor, a revolute pair is applied to the first pair in the added chain. Then, a condition on the crank angle θa,1 and θa,2 , when the output point passes through pr,1 and pr,2 respectively, was specified as θa,1  θa,2 , and added into constraint conditions of the optimization. The optimization problem was solved with the interior point method. The synthesized linkage is shown in Fig. 7. Both the selected second and the third pair in the added chain were prismatic pairs. Finally, the synthesized mechanism was evaluated. Errors between designated and output points were 3.2 × 10−3 mm and 4.8 × 10−3 mm at pr,1 and pr,2 , respectively. The OTI shown in Fig. 6 shows that the linkage does not have singular positions. In addition, the operation test of the linkage in a 3D-CAD was performed and successful as shown in Fig. 7. Therefore, an example was able to be synthesized with the proposed methodology.

6

Conclusion

In this paper, a semi-automatic method to synthesize both structure and dimension of a closed-loop spatial path-generator was proposed. The obtained results are shown as follows. (1) A general kinematic model of the linkage was constructed by parameterizing kinematic constraints of lower pairs as sets of binary values. (2) An optimal synthesis method to derive the suitable number of links, selection of kinematic pairs, and optimal dimensions to generate the desired trajectory precisely was proposed. (3) A new transmission index which can be calculated easily was proposed and included in the objective function of the optimization. (4) An example of the spatial-path generator was able to be synthesized successfully with the proposed methodology. In this method, the OTI sometimes can be zero between two designated points. Although this problem can be solved if many output positions are designated, the calculation time becomes large. To solve this problem is future work. Acknowledgements. This work was performed at Institute of Mechanism Theory, Machine Dynamics and Robotics in RWTH Aachen University in Germany. The authors are very grateful to researchers at the institute for informative discussions.

References 1. Liu, Y., McPhee, J.: Automated type synthesis of planar mechanisms using numeric optimization with genetic algorithms. J. Mech. Des. 127(5), 910–916 (2005)

A Semi-automatic Type Synthesis of a Closed-Loop Spatial Path-Generator

263

2. Kim, Y.Y., et al.: Automatic synthesis of a planar linkage mechanism with revolute joints by using spring-connected rigid block models. J. Mech. Des. 129(9), 930–940 (2007) 3. Yoon, G.H., Heo, J.C.: Constraint force design method for topology optimization of planar rigid-body mechanisms. Comput.-Aided Des. 44, 1277–1296 (2016) 4. Kang, S.W., Kim, S.H., Kim, Y.Y.: Topology optimization of planar linkage systems involving general joint types. Mech. Mach. Theory 104, 130–160 (2016) 5. Liu, X.-J., Wu, C., Wang, J.W.: A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators. J. Mech. Robot. 4, 1–10 (2012) 6. Liu, H., et al.: A generalized approach for computing the transmission index of parallel mechanisms. Mech. Mach. Theory 74, 245–256 (2014)

Walking Robot Leg Design Based on Translatory Straight-Line Generator Sayat Ibrayev1 , Nutpulla Jamalov1,2 , Amandyk Tuleshov1 , Assylbek Jomartov1(B) , Aidos Ibrayev1,2 , Aziz Kamal1,2 , Arman Ibrayeva1,2 , and Kuatbay Bissembayev1 1 Institute Mechanics and Mechanical Engineering, Almaty 050010, Kazakhstan

[email protected] 2 Al-Farabi Kazakh National University, Almaty 050010, Kazakhstan

Abstract. The planar straight-line generating linkages are of great interest for walking robot propel, especially to reduce power consumption and simplify control. Optimal design of six-link leg linkage is proposed with unlimited foot adaptation on terrain irregularities due to rectilinear and translatory motion of the output link, referred to as shin-link. The analytical method of synthesis is proposed based on least-square approximation. Due to minimizing directly the deviation from the desired output motion, the method does not suffer from so called branching defect and allows synthesizing the mechanism with desired transmission angle. Multicriteria optimization attaining both the best accuracy and transmission angle is carried out and leg-linkage with maximal value of duration ratio of support and transference phases of the leg step cycle is presented. Keywords: Walking robot · Six-link leg · Lambda mechanism

1 Introduction Designing of walking robot leg-linkage with foot center tracing along straight-line has certain advantages, considering first of all energy efficiency and simplified control [1–8]. Using of straight-line generators for walking vehicle propel still remains a productive idea since 1878 when P. Chebyshev proposed “lambda mechanism”. A numerous of studies are devoted to dimensional synthesis of one D.O.F. multi-link leg mechanisms for the last few years [4–13, et al.]. The methods of synthesis of adjustable path generators were applied for leg design [13–17]. However the main disadvantage of proposed mechanisms is poor adaptation range of leg foot on the rough terrain. Meanwhile the problem of adaptation could be solved easily by synthesizing the mechanism with translatory motion of the output link, so that not only specified coupler point, but any point of the output link traces the same straight-line [4, 6, 16, 17]. However the mechanisms synthesized by existing methods often have unfavorable transmission angle. Another drawback is “branching defect”; when one part of the desired motion is generated by one assembly of the mechanism, while another one is reproduced by another one. The main reason for these disadvantages is that the existing methods are based on the minimization of structural error, implying the compliance to specific geometric constraints, © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 264–271, 2021. https://doi.org/10.1007/978-3-030-58380-4_32

Walking Robot Leg Design Based on Translatory Straight-Line Generator

265

such as circle fitting (matching a circular point on a coupler) etc., instead of minimizing directly the error on the mechanism output. Therefore, synthesized mechanism often has unfavorable force transmission or even doesn’t work at all due to the break of the mechanism kinematic chain. The method of synthesis proposed in this paper doesn’t suffer from these disadvantages. The results of optimal design of six link leg linkage with straight-line and translatory motion of the output link and with unlimited adaptation range on the rough terrain are presented. The optimal step cycle parameter of the leg is attained by maximizing the duration ratio of support and transference phases and, at the same time; the best values of both the accuracy and transmission angle are achieved.

2 Six Link Leg Mechanism Structures with Translatory Output Motion Two mechanism structures with translatory motion of the output link (to be shin-link) EJ are plotted in Fig. 1. Here the basic straight-line generating mechanism ABCD is “lambda-type mechanism” of P. Chebyshev, link 5 (EJ) of the additional kinematic chain 4–5 copies the trajectory of the coupler point E due to the translational motion of this link. Thus with the additional input link 6 and input joint M we obtain the adjustable leg mechanism for multiple straight-line generation: now one can easily change foot height and adjust foot position on uneven surface. Note that the additional actuator at joint M is used just to fix foot position on the output link 5 (to fix link 6 relative to link 5), so joint M is passive on the support phase of the step cycle. On the support phase τSUP (that correspond to crank angle ϕAB change within the range ϕ0 ≤ ϕAB ≤ ϕ0 + SUP ): the leg foot F has to trace straight-line with nearly constant velocity relative to the robot chassis while the angular velocity of crank AB is constant (the relation between the input and output velocities has to be nearly linear) and, for the robot stabile motion; the duration of the support phase have to be longer than those of the transfer phase τTRANSF (when foot is off ground, that correspond to crank angle ϕAB values within the range ϕ0 + SUP ≤ ϕAB ≤ ϕ1 , where ϕ1 = ϕ0 + 360°). If crank AB angular velocity is constant, then the ratio υ = SUP /TRANSF is equal to duration ratio of two phases. This ratio is referred to as the parameter of step cycle. In classical Chebyshev’s lambda mechanism with SUP = 184°, TRANSF = 176° this ratio υ is equal to 1,045. The prototype of the legged robot horizontal propulsion mechanism was designed in Institute of Mechanics and Mechanical Engineering of Kazakhstan Ministry of Sciences (Fig. 2). For stabile motion of the vehicle, two legs should occur alternately in the support/transference phases while the crank angles of these mechanisms are shifted by 184° (Fig. 2). To overlap support phases of these two mechanisms, support phase should be longer than transference phase: SUP  180° and thus the parameter υ ≥ 1 should be as greater as possible. However, according to the previous studies [16], increasing this parameter leads to poor solutions in terms of both accuracy and transmission angle. A new method of synthesis proposed in the next section does not have these inherent drawbacks due to minimizing the deviation of the output motion from the desired/prescribed one directly, i.e. instead of minimizing the residual error, derived from geometric constraints, such as circle fitting, etc.

266

S. Ibrayev et al.

Fig. 1. P. Chebyshev’s “lambda mechanism” (a) and six link leg linkages with translational output motion (b), (c)

Fig. 2. Prototype of the legged robot horizontal propulsion mechanism

Fig. 3. Straight-line generator synthesis scheme

3 Approximate Synthesis of Path Generator by Minimizing Directly the Output Accuracy Let us consider four-bar linkage ABCD that is described by linkage dimensions XA = YA = 0, XD , YD , rAB , lBC , lCD (Fig. 3), where XA , YA , XD , YD are the absolute coordinates of frame pivots A and D relative to the absolute reference frame OXY . Crank angle ϕAB is varied within the range ϕ0 ≤ ϕAB ≤ ϕ0 + SUP as ϕi = ϕ0 + SUP (i − 1)/(N − 1), i = 1, . . . , N ,

Walking Robot Leg Design Based on Translatory Straight-Line Generator

267

and for each crank angle ϕAB = ϕi position analysis is supposed to be carried out, thus N positions of the coupler BC are supposed to be determined. If Bxy is the local moveable reference frame fixed on the coupler BC, with the axis Bx lying along the link BC, so we know the coordinates XBi , YBi of the origin B and inclination angles βi = βBC . Now the coupler point (end - effector) E with the local coordinates xE , yE relative to Bxy is sought, that trace through N prescribed points E1∗ , . . . , EN∗ with the given absolute coordinates Xi∗ , Yi∗ , (i = 1, . . . , N ). The absolute coordinates XEi , YEi of joint E are given by  XEi = XBi + xE cosβi − yE sinβi (1) YEi = YBi + xE sinβi + yE cosβi The designing task is formulated as the following minimization problem N δ= δi2 ⇒ min xE ,yE ,Sx ,Sy

i=1

(2)

where δ is design error  - thedistance between actual positions Ei and the desired ones Ei∗ , shifted by vector Sx , Sy  2  2  2 δi2 ≡ Ei Ei∗ 2 = XEi − Xi∗ − Sx + YEi − Yi∗ − Sy . (3) The design parameters are x1 = xE , x2 = yE , x3 = Sx , x4 = Sy , where Sx , Sy are shifting of the prescribed points Ei∗ (note that the crank pivot A coordinates were specified above to be X A = Y A = 0). From Eq. (1), δ is written as follows δ=

N i=1



XBi + x1 cosβi − x2 sinβi − Xi∗ − x3

2

2

 + YBi + x1 sinβi + x2 cosβi − Yi∗ − x4

(4)

Here XBi , YBi , βi , Xi∗ , Yi∗ are known, the necessary condition of minimum ∂δ/∂xk = 0, k = 1, . . . , 4, yield 4 linear equations with 4 unknowns AX = b

(5)

The matrix A can be written in the form A = BT B ⎡

C1 ⎢ C2 ⎢ B=⎢ . ⎣ ..

(6)

⎤ ⎥ ⎥ ⎥ ⎦

(7)

CN where,



 cosβi −sinβi −1 0 Ci = . sinβi cosβi 0 −1

Thus the matrix A is non-negatively determined and it’s all main minors as well, hence linear Eq. (5) yield single solution. It should be noted that the main advantage of this technique is that there is no branching defect while design and, moreover, the designer can specify the desired transmission angle.

268

S. Ibrayev et al.

4 Testing the Results To test the method proposed in previous sections let us use well known Chebyshev’s lambda mechanism (Fig. 1a) with the following link length: rAB = 0.5, lBC = 0.625, lCD = 0.625, frame pivot coordinates XA = 0, YA = 0, XD = −0.5, YD = 0, 0 = 90°. initial crank angle ϕAB With these parameters the coupler point P generates approximately horizontal straight-line, specified by points P1 (2, 2) and PN (0, 2). Step cycle parameters of this mechanism are SUP = 184°, TRANSF = 176° and υ = SUP /TRANSF = 1.045. The desired motion is given by N prescribed equidistant points (Xi , Yi ), i = 1, . . . , N on horizontal straight-line segment of length L = 1 determined by Y0 = 0, X0 = 0, XN = 1, as follows  Yi = Y0 , Xi = X0 + (XN − X0 ) Ni−1 −1 So the coupler point has to trace equidistant horizontal points (Xi , Yi ) while the crank angle ϕi is changed by constant increment ϕ = ϕi − ϕi−1 = sup /(N − 1), thus the relation between input and output velocities (coupler point horizontal velocity and crank angular velocity) will be nearly linear. The mechanism synthesis parameters are determined to be: x1 = xE = 1.23218, x2 = yE = −0.02180, x3 = Sx = 2.04035, x4 = Sy = 1.95574 Minimum of the Euclidean norm (2) is achieved with these  parameters. The real   Ei E ∗ 2 = 0.002, accuracy (maximal error on the output) achieved is ε = max i 2 i=1,...,N

i.e. 0.2%, trajectory height YP0 = −0.9827.

5 Optimal Leg Design The output motion is straight-line segment of unit length, given by prescribed N points given by Y0 = 0, X0 = 0, XN = 1. The support phase parameter is given as SUP = 221°, the ratio υ = SUP /TRANSF is given to be υ = 1.57. The variable parameters are varied using Sobol-Statnikov’s random LPτ - sequences [18]. For each random value of these dimensions: the worse transmission angle μe = min(|μi |, 180 − |μi |) is T calculated; synthesis parameters  X = [x1 , . . . , x4 ] are determined by solving Eqs. (5); 2 Ei E ∗  is calculated; the results are entered in the trial the accuracy ε = max i=1,...,N

i

2

table. In first stage the trial tables truncated by the accuracy δ ≺ 0.006 (0.6% from step length) and transmission angle μe  22° are studied. The mechanism dimensions with best accuracy and the best transmission angle are presented in Table 1 and 2 correspondingly. The final mechanism is plotted in Fig. 4, the mechanism structural scheme correspond to the scheme Fig. 1c. The adjusting joint M is not shown in this figure, but the adaptation is ensured due to straight-line and translational motion of the output link EF.

Walking Robot Leg Design Based on Translatory Straight-Line Generator

269

Table 1. The mechanism dimensions with best accuracy. rAB [m]

lBC [m] lCD [m]

0,2335

0,4926

0,5037

x1 = xE

x2 = yE x3 = Sx

0,9568

0,0271

XD [m]

YD [m] ϕ0 [deg]

2,0543

−0,4243 0,0301

245

x4 = Sy 1,8465

δ

μe

0,0049

22,2

Table 2. The mechanism dimensions with best transmission angle. rAB [m]

lBC [m] lCD [m]

0,2445

0,4867

0,5002

x1 = xE

x2 = yE x3 = Sx

0,9230

0,1457

XD [m]

YD [m] ϕ0 [deg]

1,7207

−0,4531 0,0925

238

x4 = Sy 1,8693

δ

μe

0,0060

25,5

Fig. 4. Designed leg mechanism.

6 Conclusions Optimal design of six-link leg linkage is proposed for walking robot horizontal propel with unlimited range of foot adaptation on terrain irregularities due to rectilinear and translatory motion of the output link. This is important for the reasons of energy efficiency and simplification of control. The analytical method of synthesis is proposed based on least-square approximation. Due to minimizing directly the deviation from the desired output motion, the method does not suffer from so-called branching defect and allows

270

S. Ibrayev et al.

synthesizing the mechanism with desired transmission angle and accuracy range. The step cycle parameter υ = SUP /TRANSF is increased up to 1.59, support phase angle SUP is increased up to 221°, instead of the values in prototype υ = 1.045, SUP = 184°.

References 1. Song, S.M., Vohnout, V.J., Waldron, K.J., Kinzel, G.L.: Computer-aided design of a leg for an energy efficient walking machine. Mech. Mach. Theory 19(1), 17–24 (1984) 2. Ryan, A.D., Hunt, K.H.: Adjustable straight-line linkages - possible legged-vehicle applications. Trans. ASME: J. Mech. Transm. Autom. Des. 107, 256–261 (1985) 3. Song, S.M., Waldron, K.J.: Machines that Walk: The Adaptive Suspension Vehicle. MIT Press, Cambridge (1989) 4. Umnov, N.V.: Peculiarities of using legged mechanisms in non-traditional vehicles (chap. 11). In: Encyclopedia of Machinery, Part 2, vol. 1. Mashinostroenie Press, Moskow (1996). (in Russian) 5. Xu, K., Liu, H., Zhu, X., Song, Y.: Kinematic analysis of a novel planar six-bar bionic leg. In: Proceedings of 15th IFToMM World Congress on Mechanism and Machine Science, Krakow, pp. 13–22 (2019) 6. Briskin, E.S., et al.: Mathematical modelling of mobile robot motion with propulsion device of discrete interacting with the support surface. In: Proceedings 9th Vienna International Conference on Mathematical Modelling, MATHMOD 2018, Vienna, Austria, pp. 259–264 (2018) 7. Komoda, K., Wagatsuma, H.: Energy-efficiency comparisons and multibody dynamics analysis of legged robots with different closed-loop mechanisms. Multibody Syst. Dyn. 40(2), 123–153 (2017) 8. Özgun, S., Ceccarelli, M.: Design and optimization of a walking over-constrained mechanism. In: Proceedings of 15th IFToMM World Congress on Mechanism and Machine Science, Krakow, pp. 681–687 (2019) 9. Kim, H., Jung, M., Shin, J., Seo, T.: Optimal design of Klann-linkage based walking mechanism for amphibious locomotion on water and ground. J. Inst. Control Robot. Syst. 20(9), 936–941 (2014) 10. Plecnik, M., McCarthy, J.: Design of Stephenson linkages that wide a point along a specified trajectory. In: New Trends in Mechanism and Machine Science, pp. 273–280. Springer, Netherlands (2016) 11. Selvi, Ö., Samet, Y.: Design and dimensional optimization of a novel walking mechanism with firefly algorithm. In: International Workshop on Computational Kinematics. Springer, Cham (2017) 12. Wang, C.Y., Hou, J.H.: Analysis and applications of Theo Jansen’s linkage mechanism-Theo Jansen’s linkage mechanism on kinetic architecture. In: Proceedings of the 23rd CAADRIA Conference, pp. 359–368. Tsinghua University, Beijing (2018) 13. Funabashi, H.: Adjustable mechanisms with variable crank length. In: Proceedings of the 7-th World Congress, Sevilla, Spain, pp. 65–68 (1987) 14. Chanekar, P.V., Ghosal, A.: Optimal synthesis of adjustable planar four-bar crank-rocker type mechanisms for approximate multi-path generation. Mech. Mach. Theory 69, 263–277 (2013) 15. Peng, C., Sodhi, R.S.: Optimal synthesis of adjustable mechanisms generating multi-phase approximate paths. Mech. Mach. Theory 45(7), 989–996 (2010) 16. Ibrayev, S.M.: Synthesis of walking vehicles leg mechanisms with decoupled foot-point motion. In: Proceedings of Tenth IFToMM World Congress on TMM, Oulu, Finland, pp. 1188–1195 (1999)

Walking Robot Leg Design Based on Translatory Straight-Line Generator

271

17. Ibrayev, S.M.: Approximate synthesis of planar Cartesian manipulators with parallel structures. Mach. Mech. Theory 37, 877–894 (2002) 18. Statnikov, R.B.: Multicriteria Design. Optimization and Identification. Kluwer Academic Publishers, Dordrecht (1999)

Developing a Flexible Segment Unit for Redundant-DOF Manipulator Using Bending Type Pneumatic Artificial Muscle Hiroki Tomori1(B) , Tomohiro Koyama1 , Hiromitsu Nishikata1 , Akinori Hayasaka1 , and Ikumi Suzuki2 1 Yamagata University, Yamagata, Japan

[email protected] 2 Nagasaki University, Nagasaki, Japan

Abstract. Robots for human are required to have safety while working with them. Therefore, we focused to a pneumatic artificial muscle which has intrinsic flexibility. In this paper, we developed a bending-type pneumatic rubber artificial muscle (BPAM) and a unit driven by 3 BPAMs. We are aiming to develop redundant degree of freedom manipulator by connecting a number of units. We also conducted load characteristic experiment of the unit. Experimental result showed that the unit can drive omnidirectionally as we proposed. However, the unit has uneven output characteristics because these BPAMs were made by hand, individual differences in characteristics of BPAMs were occurred for example thickness of rubber tube and a little misregistration of aramid fiber in axial length. Then, the unit was controlled by neural network learning. Trajectory tracking experiment result showed that actual trajectory was off-center from desired trajectory. Nonetheless, the unit reproduced tendency of trajectory, and we consider that the trajectory control can be improved by adding attitude feedback control. Keywords: Artificial muscle · Pneumatic · Actuator · Manipulator · Rubber

1 Introduction Recently, robots working near by people are increasing. These robots are required to be safe in case of collision with around people and objects. In addition, their working field is almost always complicated area such as living environment of people. Although robots driven by stiff actuators can avoid collision and reduce damage of each other using depth, force, and torque sensors [1, 2], it difficult to soften any unintended and instantaneous contact, and unexpected collisions are difficult to prevent in complex environments. Therefore, we focused on soft actuators. These actuators are typically constructed from soft materials such as rubber, gel, and polymers. Though flexible actuators tend to be less powerful, less accurate, and less responsive than stiff actuators, their flexibility is advantageous in terms of their constructive safety [3]. Suzumori developed an arm consisting of an air balloon to achieve height safety [4] and a pipe inspection robot using flexible microactuators, which can adapt to a narrow and complex space without © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 272–279, 2021. https://doi.org/10.1007/978-3-030-58380-4_33

Developing a Flexible Segment Unit for Redundant-DOF Manipulator

273

complicated control [5]. Ansari developed a manipulator using McKibben actuators for the personal care of elder people [6]. Hawkes developed a soft robot that can navigate in a narrow space by growing [7]. Such a behavior is possible because these robots have a soft construction. In addition, soft skins that contribute to robots increasing environment adaptability [8] and grasping ability [9] were developed. In this manner, soft robots have advantages in terms of not only safety for human and the surrounding environment, but also in terms of performance such as the adaptability toward complicated and indeterminate environments, and in reducing calculation costs in control and the grasping ability of the manipulator. In this study, we aim to develop redundant degree of freedom (DOF) manipulator driven by bending-type pneumatic rubber artificial muscle. This actuator consists of fibers and a flexible membrane like rubber. It actuates via deformation (expansion and contraction), upon the application of air pressure. The actuator has no sliding or moving parts, such as a pneumatic cylinder [10], which means some advantages such as dust proofing, high scalability, and low noise. Also, its working fluid (air) is environmentand user-friendly. Thus, its high safety encourages the application of robots toward working nearby humans, for example, in the rehabilitation field [11, 12]. Furthermore, the pneumatic actuator’s lightness and a high output relative to other soft actuators are of advantages to increase degree of freedom of the manipulator. This manipulator is assembled by connecting independent bending units in series (Fig. 1). And each unit is driven by artificial muscles. In this paper, we developed the bending unit and control it. Especially, to address hysteresis and nonlinear characteristics of artificial muscles, we applied neural network learning to determine input air pressure.

Fig. 1. A redundant DOF manipulator.

2 Bending-Type Pneumatic Rubber Artificial Muscle (BPAM) Photographs of the BPAM are displayed in Fig. 2 and the specifications of the BPAM are listed in Table 1. This actuator consists of a rubber tube and an aramid fiber, and bends by applying air pressure. Here, the drive principle of BPAM is explained with Fig. 3. In Fig. 3, a rubber tube (A) expands in all directions (B) with air pressure by Pascal’s principle. Therefore, the aramid fiber rolled in the rubber tube without space restraints

274

H. Tomori et al.

expands in a radial direction (C). In this case, the tube extends in only the axial direction by applying air pressure (D). In addition, an aramid fiber of 8 mm width was utilized to limit the axial length of a part of the tube walls (E). Due to this mechanism, the difference of extension in the tube upon the application of air pressure produces bending of the BPAM in the direction where the axial length is constrained (F).

Fig. 2. A bending-type pneumatic artificial muscle: (left) air is released, and (right) air is applied.

Table 1. Bending-type pneumatic artificial muscles (BPAM) specifications. Parameter

Specification

Length (mm)

100

Outer diameter (mm) 21 Inner diameter (mm) 16 Weight (kg)

0.05

Fig. 3. Mechanism of bending-type pneumatic artificial muscles (BPAM).

Developing a Flexible Segment Unit for Redundant-DOF Manipulator

275

3 Flexible Bending Unit 3.1 Mechanism and Drive Principle of the Unit Figure 4 shows a flexible bending unit. This unit consists of three BPAMs, and each BPAM is arranged to bend inward. Due to this structure, this unit can bend in resultant force direction by activating more than one BPAMs. We consider that activating less than two BPAMs can control the unit in omnidirectionally, and balancing all three BPAM produces stiffness control. In this paper, only attitude of the unit was controlled by two or less BPAMs. Air from compressor is controlled by solenoid valves connecting to each BPAM like Fig. 5. In addition, the unit has a 6-axis IMU on Arduino nano 33 IOT to detect attitude of the unit bending. Then, we defined a coordinate of the unit as Fig. 6.

Arduino nano 33 IOT (6-axis IMU)

Fig. 4. A flexible bending unit.

Air compressor

PC Micro computer

Solenoid valve

6-axis IMU

Fig. 5. Control system of the flexible bending unit.

Then, we drive this unit on a trial basis to grab basic property. In this test, 0.05 MPa and 0.1 MPa of air pressure applied to one or two BPAMs, and load of 0 g to 200 g were applied to tip of the unit. Here, Fig. 7 shows a result of driving. Although these

276

H. Tomori et al. Y axis Pitch BPAM 3

BPAM 2

X axis Roll

BPAM 1

Fig. 6. Coordinate of the unit (View from sensor side).

plots were expected to be symmetrical, hexagonal shape formed by plots were distorted. Because these BPAMs were made by hand, individual differences in characteristics of BPAMs were occurred for example thickness of rubber tube and a little misregistration of aramid fiber in axial length. Nonetheless, the unit can drive omnidirectionally as we proposed. 0.1MPa, 0g 0.05MPa, 0g

20

0.1MPa, 100g 0.05MPa, 100g

BPAM 2 and 3

BPAM 2

15

0.1MPa, 200g 0.05MPa, 200g

10

BPAM 3

Pitch[deg]

5 0 -5

BPAM 1 and 2

-10

BPAM 1 and 3

-15 BPAM 1

-20 -20

-15

-10

-5

0 5 Roll[deg]

10

Fig. 7. Results of unit driving test.

15

20

Developing a Flexible Segment Unit for Redundant-DOF Manipulator

277

4 Trajectory Tracking Experiment Using Neural-Network Learning To control this unit, it needs to determine pressure inputs for each BPAM. However, this unit has some factors that make control complicated: output property and hysteresis of BPAM, interaction between BPAMs, individual difference, etc. We thought that if the unit can be controlled its attitude roughly, feedback control will be applied to the controller for fine adjustment. In this chapter, we aim to control the unit by neural network learning as first step of control system design. The aim of neural network is to predict the air pressures for each BPAM given the pitch and the roll for the unit to drive. We construct a neural network architecture as shown in Fig. 8. The nodes in the neural network is a dense, fully-connected. The layers in the network consists of an input, a hidden and an output layer. The each layer consists of 7, 300 and 3 nodes, respectively. We applied MLPRegressor in Scikit-learn machine learning library to implement our neural network model. The learning parameters were set as lbfgs for optimizer and alpha = 0.0001 for L2 penalty parameter. We collected 2120 data by driving the unit randomly. Each data contains input and output. The output is three air pressure values for each BPAM. As for the input, we include not only pitch and roll, but also the previous state of pitch, roll and three air pressure values. It was important to include previous state information as input because the state of the unit heavily depends on the previous state. To train and test the neural network, we applied leave-one-out cross validation.

Input Layer

Hidden Layer

Output Layer

Fig. 8. Illustration of neural network architecture.

Figure 9 and Fig. 10 show results of trajectory tracking experiment. Although actual trajectory moved from desired trajectory in same direction in whole, the unit reproduced

278

H. Tomori et al.

tendency of trajectory. We consider that the trajectory control can be improved by adding attitude feedback control.

Fig. 9. Result of trajectory tracking experiment (roll angle).

Fig. 10. Result of trajectory tracking experiment (pitch angle).

5 Conclusion To develop human friendly redundant DOF manipulator, we developed the flexible segment unit using three BPAMs. Then, basic property experiment result showed that the

Developing a Flexible Segment Unit for Redundant-DOF Manipulator

279

unit has uneven output characteristics because these BPAMs were made by hand, individual differences in characteristics of BPAMs were occurred for example thickness of rubber tube and a little misregistration of aramid fiber in axial length. Nonetheless, the unit can drive omnidirectionally as we proposed. Next, the unit was controlled by neural network learning. Trajectory tracking experiment result showed that actual trajectory was off-center from desired trajectory. However, the unit reproduced tendency of trajectory, and we consider that the trajectory control can be improved by adding attitude feedback control. In future works, we plan to add feedback control and assemble a redundant manipulator by connecting units serially. Then, we aim to propose motion planning method of the manipulator.

References 1. Ragaglia, M., Zanchettin, A.M., Rocco, P.: Trajectory generation algorithm for safe humanrobot collaboration based on multiple depth sensor measurements. Mechatronics 55, 267–281 (2018) 2. Sim, O., Oh, J., Lee, K., Oh, J.H.: Collision detection and safe reaction algorithm for nonbackdrivable manipulator with single force/torque sensor. J. Intell. Robot. Syst. 91(3–4), 403–412 (2018) 3. Yang, H., Xu, M., Li, W., Zhang, S.: Design and Implementation of a soft robotic arm driven by SMA coils. IEEE Trans. Ind. Electron. 66, 6108–6116 (2019) 4. Takeichi, M., Suzumori, K., Endo, G., Nabae, H.: Development of a 20-m-long Giacometti arm with balloon body based on kinematic model with air resistance. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems, Vancouver, BC, Canada, pp. 2710–2716 (2017) 5. Suzumori, K., Abe, A.: Applying flexible microactuators to pipeline inspection robots. In: Proceedings of the IMACS/SICE International Symposium on Robotics, Mechatronics and Manufacturing Systems, Tempe, AZ, USA, pp. 515–520 (1993) 6. Ansari, Y., Manti, M., Falotico, E., Mollard, Y., Cianchetti, M., Laschi, C.: Towards the development of a soft manipulator as an assistive robot for personal care of elderly people. Int. J. Adv. Robot. Syst. 14, 1–17 (2017) 7. Hawkes, E.W., Blumenschein, L.H., Greer, J.D., Okamura, A.M.: A soft robot that navigates its environment through growth. Sci. Robot. 2, 1–7 (2017). eaan3028 8. Digumarti, K.M., Conn, A.T., Rossiter, J.: Pellicular morphing surfaces for soft robots. IEEE Robot. Autom. Lett. 4, 2304–2309 (2019) 9. Bolson, N., Singh, D., Lube, V., Lubineau, G.: All-polymer based polymorph skin with controllable surface texture. Smart Mater. Struct. 28, 075011 (2019) 10. Sarosi, J.: Elimination of the hysteresis effect of PAM actuator: modelling and experimental studies. Tech. Gaz. 22, 1489–1494 (2015) 11. Deaconescu, T., Deaconescu, A.: Pneumatic muscle actuated isokinetic equipment for the rehabilitation of patients with disabilities of the bearing joints. In: Proceedings of the International MultiConference of Engineers and Computer Scientists, Hong Kong, China, vol. 2, pp. 1823–1827 (2009) 12. Li, X., Xia, H., Guan, T.: Development of legs rehabilitation exercise system driven by pneumatic muscle actuator. In: Proceedings of the 2008 2nd International Conference on Bioinformatics and Biomedical Engineering, Shanghai, China, pp. 1309–1311 (2008)

Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism Vu Linh Nguyen1 , Chin-Hsing Kuo2(B) , and Chyi-Yeu Lin1 1 National Taiwan University of Science and Technology, Taipei 106, Taiwan 2 University of Wollongong, Wollongong, NSW 2522, Australia

[email protected]

Abstract. This paper presents a design concept for the gravity compensation of the renowned Delta parallel robot. The design is constructed by using three gear-spring modules, each being installed on the proximal link of each leg of the robot. This implementation can allow the balancing design to merely request a relatively small assembly space on the robot, thereby avoiding mechanical interference between the supplemental structures and robot links during operation. The spring design is realized by an analytical approximation to perfect balancing. A design example for an industrial Delta robot, namely FANUC M-3iA/12H, is also given. It is shown that the studied robot can theoretically demonstrate a 41.4% total motor torque reduction during a pick-and-place operation under a 10-kg load of the moving platform. Keywords: Gravity compensation · Static balancing · Robot performance

1 Introduction Gravity compensation for robot manipulators can allow the robot to reduce the driving torques and actuator size, thereby increasing the robot performance, safety, and energy efficiency [1, 2]. Great efforts have been paid by researchers for improving the performance of the renowned Delta parallel robot [3]; however, gravity compensation methods for such robot structures are still limited. There is no convergence on gravity-balancing solutions so far for the Delta robot, though many balancing designs for serial or parallel robots exist [4, 5]. Gravity compensation for the Delta parallel robot can be realized by attaching energy storage elements, mainly counterweights and springs, to the robot structure. This implementation can allow canceling the variation of the gravitational energy of the robot during operation. Conventional balancing designs for the Delta robots usually suffer from a significant increase in the total robot weight and inertia when using counterweights, or structural complexity when using springs. For instance, Baradat et al. [6] presented a balancing mechanism constructed by a multi-loop pantograph with a counterweight for the Delta robot. This balancing design could provide a high performance of gravity compensation with the aid of parametric optimization. However, such a structure may cause mechanical interference associated with the robot links, and raise the total weight, © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 280–287, 2021. https://doi.org/10.1007/978-3-030-58380-4_34

Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism

281

inertia, and bulkiness of the robot. Wijk and Herder [7] presented several designs of force-balanced Delta robots. This study suggested that such a robot structure can be balanced by using three counterweights and two additional links. However, a ratio of the balancing mass over the initial mass of 4.4 could be received from this design. Dam et al. [8] proposed a balancing design using zero-free-length (ZFL) springs. Although the method could introduce a simple balancing design without auxiliary links, the attachment of springs closed by the moving platform can increase the inertial forces on the robot. Simionescu et al. [9] introduced two perfect/approximate balancing designs for Delta structure. While the perfect design is somewhat complex and bulky, the approximate one is simpler and more practical due to the decreased number of springs. Nevertheless, the approximate design may significantly limit the balancing workspace. It should be noticed that the use of ZFL springs in [8, 9] can result in practical inconvenience and friction issues, as discussed in [1]. In this paper, a new design concept for the gravity compensation of the Delta parallel robot using the gear-spring modules (GSMs) is presented. The GSM has been successfully proposed for the gravity compensation of serial robots [10]. In the Delta balancing design, a GSM is installed on the proximal link of each leg of the robot. This design strategy can bring the robot with a couple of advantages, such as provide a compact mechanical structure, maintain the robot workspace without mechanical interference between the added parts and robot links, and the design does not request a parametric optimization. Moreover, the performance of the proposed balancing design is demonstrated through a numerical example with the FANUC M-3iA/12H Delta robot. In what follows, the description of the balancing robot is given in Sect. 2. Then, the design for gravity compensation is explained in Sect. 3. Next, Sect. 4 presents the design example with the FANUC Delta robot. Lastly, Sect. 5 concludes the present paper.

Fixed base Actuation joint

Connecting rod

Link ai

Spring

Slider

Spur gears

Gear-spring module (GSM i)

Link bi1 Link bi2 Moving platform

Fig. 1. The gravity-compensated GSM-based Delta parallel robot.

282

V. L. Nguyen et al.

2 Description of the Balancing Robot The gravity compensation design of the Delta parallel robot is shown in Fig. 1. In this design, three identical GSMs are installed on the proximal links of the three legs of the robot, respectively. As explained in [10], the GSM is essentially a five-bar gearslider mechanism with a compression spring for producing an elastic (spring) energy for gravity compensation. The geometry of the Delta robot is explained in Fig. 2. Here, the coordinate systems x0 y0 z0 , xai yai zai , xbi ybi zbi (i = 1, 2, 3), and xE yE zE are attached to the center O of the fixed base, joint Di , joint Pi , and the center E of the moving platform, respectively. Let l a and l b denote the lengths of link ai (la = Di Pi ) and link bij (l b = Pi Qi = Pij Qij , j = 1, 2), respectively. Let mai , mbij , and mE be the masses of link ai (including the mass of GSM i), link bij (j = 1, 2, 3, and 4), and the moving platform (including a payload), which are located at the geometrical centers C ai , C bij , and E, respectively. As shown in Fig. 2(b), all the link masses mbij of parallelogram Pi1 Pi2 Qi2 Qi1 can be expressed by an equivalent mass mbi located at the geometrical center C bi of the parallelogram. Let θ 1i , θ 2i , and θ 3i be the joint angle of link ai , the angle between link ai and the projection of link bij (j = 1, 2) on the xai yai plane, and the angle between link bij (j = 1, 2) and the zbi axis, respectively. y0

x0 Fixed base

O z0

p0i zai

yai

yai Di xai

Tm/si

θ1i

zbi

Cai la

Pi2

Pi

yai

xbi

xE

E mE zE

yE pEi Qi1 (a)

Pi

sbi

ybi

zbi

xbi

θ3i

mai

Pi1

Pi2 Pi

θ2i

p

Cbi

Cbi Link bi1

Di

Cai

Link bi3

θ2i

Moving platform

zai

xai

sai

ybi Pi1

Di

θ1i

z0

Link ai

Cai maig Gravity

p0i

O

Cbi

mbig

mbi

lb Fi Qi

Link bi2

Qi2 Link bi4

E zE

pEi

Qi

Qi1

Front view

Qi

Qi2

Side view

(b)

Fig. 2. The geometry of the Delta robot: (a) the leg geometry and (b) descriptions of the joint angles and the equivalent mass of the parallelogram.

The geometry of the GSM is depicted in Fig. 3. In this figure, GSM i is installed on link ai of leg i of the robot. The first gear of GSM i (gear 1i) is centered at the revolute joint Di and fixed to the robot base (see Fig. 3(a)). The second gear (gear 2i) is fixed to link ai and is driven to compress the spring k i when link ai rotates clockwise. Let T si and T 2i represent the spring torques on link ai and gear 2i, respectively. Here, the initial configuration of GSM i, i.e., the spring k i is with its free length, is defined when link ai

Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism

283

aligns with the yai axis or θ 1i = –π /2 (see Fig. 3(b)). At this configuration, the angle ψ i between vectors Hi Di and Hi Ai can stand for the installation angle of GSM i. Initial configuration

yai

Fixed base

Fri

Spring i

Gravity

αi

xai

Ai

Hi

φci

Tsi T2i

Slider i

Bi

θ1i

Di

Connecting rod i

Bi

ψi

Rotating link i

φ0i ψi

r3i

Link ai

r2i dsi

Cai

ki

Gear 2i

Hi

Ai

Gear 1i

maig

r2ai

Link bi1

(a)

yai

r1i Di

xai

(b)

Fig. 3. The geometry of the gear-spring module (GSM): (a) the installation scheme on the robot and (b) the initial configuration.

3 Design for Gravity Compensation 3.1 Torque Calculation From Fig. 2, the gravitational torque T wi at joint Di induced by the weights of the robot links and moving platform can be determined as: Twi = rai × mai g + rbi × mbi g + rci × Fi 

(1)

where rai , rbi , and rci represent the position vectors of the geometrical centers C ai , C bi , and joint Qi , respectively; ||.|| denotes the Euclidean norm of a vector; Fi = (mE g)/3 is a force applied at joint Qi due to the weight of the moving platform. In Eq. (1): rai = p0i + Cai − Di , rbi = p0i + Cbi − Di , rci = p0i + Qi − Di , T  Di = −p0i  cos θpi , p0i  sin θpi , 0 , T  Cai = −(p0i  + sai cos θ1i ) cos θpi , (p0i  + sai cos θ1i ) sin θpi , sai sin θ1i , T  Pi = −(p0i  + la cos θ1i ) cos θpi , (p0i  + la cos θ1i ) sin θpi , la sin θ1i , T  sbi Qi = xE − pEi  cos θpi , yE + pEi  sin θpi , zE , Cbi = Pi + (Qi − Pi ) lb

(2)

284

V. L. Nguyen et al.

In Eq. (2), p0i and pEi represent the position vectors from point O to joint Di and from point E to joint Qi , respectively; sai = Di C ai and sbi = Pi C bi ; p = [x E , yE , zE ]T and θ pi stand for the position vector of point E and the angle of vector p0i on the x0 y0 plane (θ p1 = π /6, θ p2 = 5π /6, and θ p3 = 3π /2), respectively. According to [10], the spring torque T si at joint Di caused by spring k i can be determined as:   ⎤ ⎡ θ1i + π 2 d0i + r3i cos ϕci − r2ai cos ψi + ⎥ ki r2ai ⎢ ngi ⎥ ⎢ Tsi = ⎦ ⎣

ngi r3i 2 2 − r3i + r2ai − 2r3i r2ai cos(ψi − ϕ0i )      /2 r3i cos ϕci − r2ai cos ψi + θ1i +π θ1i + π 2 ngi × sin ψi + (3) cos ϕci ngi where k i , r 2ai , r 3i , and ngi represent the spring stiffness coefficient, distance H i Ai , the length of the connecting rod Ai Bi , and the gear ratio of GSM i, respectively; d 0i and ψ i the initial spring compression and installation angle of GSM i, respectively; ϕ 0i and ϕ ci the initial and instantaneous deflection angles between the connecting rod and rotating link of GSM i, respectively. By assuming that d 0i = 0, ngi = 2, and r 3i > > r 2ai , the spring torque T si in Eq. (3) can be simplified to: Tsi =

  2 ki r2ai ki r 2 π π sin 2ψi − cos θ1i + 2ai cos 2ψi − sin θ1i 4 2 4 2

(4)

3.2 Determination of the Spring Stiffness and Installation Angles Let T m/si be the motor torque at joint Di used to support the robot under gravity and spring forces, and it can be defined as: Tm/si = Tsi − Twi

(5)

From Eq. (5), it can be seen that the perfect balancing over the robot workspace can be achieved if and only if: Tm/si = 0 for any θ1i

(6)

As explained in [10], it is practically nearly impossible to achieve the balancing condition in Eq. (6). An alternative way to deal with is to define the desired configuration C Θ where the balancing condition should be met. Such a configuration can be represented by a set of joint angles Θ 1i . It is noteworthy that the studied Delta robot possesses a symmetric architecture, and its workspace is centered around the z0 axis. Thus, we can reasonably assume that the desired configurations C Θ are the ones where the z0 and zE axes are collinear, i.e., Θ 1i = Θ 11 = Θ 12 = Θ 13 , leading to Θ 3i = π /2. From this assumption, the gravitational torque T wi in Eq. (1) at C Θ can be expressed as: Twi = (ATi + BTi cos Θ2i ) cos Θ1i − BTi sin Θ2i sin Θ1i

(7)

Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism

285

where ATi = mai gsai + (mbi g + Fi )la , BTi = mbi gsbi + Fi lb

(8)

By substituting Eqs. (4) and (7) into Eq. (5), the motor torque T m/si at the desired configurations C Θ can be expressed as:    2 ki r2ai π sin 2ψi − − ATi − BTi cos Θ2i cos Θ1i Tm/si = 4 2    2 ki r2ai π cos 2ψi − + BTi sin Θ2i sin Θ1i + (9) 4 2 By comparing Eqs. (6) and (9), the perfect balancing can be achieved at the desired configurations C Θ if two following conditions are simultaneously satisfied:  2 ki r2ai π − ATi − BTi cos Θ2i = 0 (10) sin 2ψi − 4 2  2 ki r2ai π cos 2ψi − + BTi sin Θ2i = 0 (11) 4 2 By solving Eqs. (10) and (11), the spring stiffness k i and the installation angle ψ i of GSM i can be found as: −4BTi sin Θ2i kπ   ki = 2 (12) π , ψi  = 2 (k = 0, 1, . . . , 4) r2ai cos 2ψi − 2 1 π (13) ψi = atan2(ATi + BTi cos Θ2i , −BTi sin Θ2i ) + 2 2 In summary, the perfect balancing condition in Eq. (6) can be achieved at the desired configurations C Θ if the spring stiffness k i and the installation angles ψ i of the GSMs are calculated by using Eqs. (12) and (13), respectively, under conditions d 0i = 0, ngi = 2, and r 3i >> r 2ai .

4 An Illustrative Example: FANUC M-3iA/12H Delta Robot In this session, the proposed design strategy is applied to an industrial Delta robot, namely FANUC M-3iA/12H. Here, a 10-kg weight of the moving platform is assumed to be gravity compensated. The design parameters of the robot are estimated from the datasheet available on the FANUC’s website, as listed in Table 1. The design parameters of the GSMs are listed in Table 2. Assume that the robot is to execute two pick-and-place operations characterized by two different trajectories, as shown in Fig. 4. By selecting the desired configuration Θ 1i = 0 for the balancing design, we can use Eqs. (12) and (13) to determine the spring stiffness (k i = 63 N/mm) and the installation angle (ψ i = 47.3°), respectively. Based on the given data in Tables 1 and 2, the total motor torque and joint angles of the robot for the two operations above are illustrated in Fig. 5. The results show that the use of GSMs can allow the robot to reduce the total motor torque during trajectory tracking. The peak motor torque can reduce up to 38.4% and 41.4% for the first and second operations, respectively.

286

V. L. Nguyen et al. Table 1. Design parameters (estimated) of the FANUC M-3iA/12H Delta parallel robot.

l a (m)

l b (m)

sai (m)

sbi (m)

p0i (m)

pEi (m)

mai (kg)

mbi (kg)

mE (kg)

0.28

0.92

0.12

0.46

0.3

0.1

3

1.5

10

Table 2. Design parameters of the GSMs. ngi r 1i (m) r 2i (m) r 2ai (m) r 3i (m) 2

0.02

0.04

0.04

0.18

End config.

Trajec. 2

End config.

Initial config.

Trajec. 1 Initial config.

(a)

(b)

Balanced

θ13

θ11, θ12

Total torque (N-m)

Unbalanced

Angle θ1i (°)

Angle θ1i (°)

Total torque (N-m)

Fig. 4. Trajectory examples: (a) pick-and-place operation I and (b) pick-and-place operation II.

Unbalanced Balanced

θ13

θ12 θ11

Time (s)

Time (s)

(a)

(b)

Fig. 5. Total motor torque during trajectory tracking: (a) operation I and (b) operation II.

Gravity Compensation of Delta Parallel Robot Using a Gear-Spring Mechanism

287

5 Conclusion In this paper, we have presented a gravity compensation design for the renowned Delta parallel robot using the GSMs. The balancing design is realized by an analytical approximation to perfect balancing based on the assumption of the desired configurations, where high performance of gravity compensation should be received. The design for a FANUC Delta robot for gravity compensation was then studied, and a peak motor torque reduction of 41.4% was achieved during a pick-and-place operation.

References 1. Herder, J.L.: Energy-free systems: theory, conception and design of statically balanced spring mechanisms. Ph.D. Thesis, Department of Mechanical Engineering, Delft University of Technology, Delft, The Netherlands (2001) 2. Carricato, M., Gosselin, C.: A statically balanced Gough/Stewart-type platform: conception, design, and simulation. ASME J. Mech. Robot. 1(3), 031005 (2009) 3. Kelaiaia, R.: Improving the pose accuracy of the Delta robot in machining operations. Int. J. Adv. Manuf. Technol. 91(5–8), 2205–2215 (2017) 4. Kim, H.-S., Min, J.-K., Song, J.-B.: Multiple-degree-of-freedom counterbalance robot arm based on slider-crank mechanism and bevel gear units. IEEE Trans. Robot. 32(1), 230–235 (2016) 5. Gosselin, C.M., Wang, J.: On the design of gravity-compensated six-degree-of-freedom parallel mechanisms. In: IEEE International Conference on Robotics and Automation (ICRA), Leuven, Belgium, May 16–20, vol. 3, pp. 2287–2294 (1998) 6. Baradat, C., Arakelian, V., Briot, S., Guegan, S.: Design and prototyping of a new balancing mechanism for spatial parallel manipulators. ASME J. Mech. Des. 130(7), 072305 (2008) 7. Van der Wijk, V., Herder, J.L.: Dynamic balancing of Clavel’s Delta robot, in Computational Kinematics, pp. 315–322. Springer, Heidelberg (2009) 8. Van Dam, T., Lambert, P., Herder, J.L.: Static balancing of translational parallel mechanisms. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE), August 28–31, Washington, DC, USA, pp. 883–889 (2011) 9. Simionescu, I., Ciupitu, L., Ionita, L.C.: Static balancing with elastic systems of DELTA parallel robots. Mech. Mach. Theory 87, 150–162 (2015) 10. Nguyen, V.L., Lin, C.-Y., Kuo, C.-H.: Gravity compensation design of planar articulated robotic arms using the gear-spring modules. ASME J. Mech. Robot. 12(3), 031014 (2020)

Design of the Mobile Robot Agri.q Paride Cavallone(B) , Carmen Visconte, Luca Carbonari, Andrea Botta, and Giuseppe Quaglia Politecnico di Torino, Turin, Italy [email protected]

Abstract. In this paper, an innovative UGV (Unmanned Ground Vehicle), named Agri.q, is presented. The rover is specifically designed for precision agriculture applications and is able to work in unstructured environment on irregular soil, cooperating with drones, if necessary. It is equipped with specific tools and sensors to per-form specific tasks, i.e. mapping the field, monitoring the crops and collecting soil and leaf samples. In addition, it is provided with a two degrees of freedom landing platform able to self-orient to ensure a safe drone docking or even to maximize the sunrays collection during the auto-charging phase. In this way, the rover autonomy and sustainability are increased. The functional design of the rover and the design of its actuation system are reported herein; furthermore, the first prototype is described and some preliminary results obtained during experimental tests are discussed. Keywords: Agri.q · Mobile robot · Precision agriculture · Service robotics · Sustainability

1 Introduction “Precision Agriculture is a management strategy that gathers, processes and analyses temporal, spatial and individual data and combines it with other information to support management decisions according to estimated variability for improved resource use efficiency, productivity, quality, profitability and sustainability of agricultural production.” [1]. Monitoring and analysing crops and soil is necessary to identify what to do, where and when to intervene. Several solutions aim to equip agricultural machines or commercial UGVs (Unmanned Ground Vehicles) with tools and devices to monitor, analyse both crops and soil [2, 3]. In other solutions the UAVs (Unmanned Aerial Vehicles) are equipped with cameras in order to monitoring the crops [4]. In addition, agricultural machines and commercial UGVs are equipped with specific tools or end-effector to do different actions [5, 6]. The authors have been studying and presented in the past the concept of an innovative UGV, named Agri.q, for precision agriculture [7–9]. Agri.q is a small sized electric rover designed to operate in unstructured environments and to move through the rows of vines. It is able to cooperate with drones and it is equipped with a robotic arm. Moreover, it is equipped with a solar panel in order to increase its autonomy and sustainability. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 288–296, 2021. https://doi.org/10.1007/978-3-030-58380-4_35

Design of the Mobile Robot Agri.q

289

2 Functional Design A modular approach has been employed for the system of locomotion. Agri.q is composed by four driving units: two front and two rear. Each driving unit is composed by two wheels housed on a rocker arm structure connected to the body machine using a passive joint, whose respective DOF (Degree of Freedom) is hereby called . The rocker arm can rotate about the joint to ensure a correct distribution of the normal forces on the ground. The two front driving units are connected with the front box. The front box is connected to the frame with a passive DOF θ joint that allows the front and the rear modules to exhibit a mutual yaw angle, necessary to approach curved trajectories. The two rear driving units are connected with the rear axle, constrained to the positioning mechanism with one passive DOF αF joint that ensures the compensation of transversal inclination between the front and the rear driving units. The positioning mechanism connects the rear axle and the frame with a one DOF αE active joint. The DOF αE defines the pitch angle of the frame. A landing platform covered with two solar panels is connected with the frame with a one DOF αR active roll joint. The two actuated DOFs αR and αE allow to orient the landing platform/solar panels in order to optimise the sun capture or to keep horizontal the landing platform for the drones. 2.1 Rocker Arm Mechanism The design of the Agri.q rover was driven by its agriculture application, and due to that eight wheels architecture, each couple supported by a rocker, has been adopted. This solution permits to distribute the static and dynamic forces acting on the ground over a wide contact surface, as occurs using a track systems, see Fig. 1a). As a result, the vehicle itself is prevented from sinking, or getting stuck, into soft terrain and, at the same time, soil compaction is avoided. Nonetheless, the whole traction efficiency remains similar to that of a four-wheeled rover. A further advantage comes from the capability of the rocker to act like a filter with respect to the oscillations imposed by the ground. Figure 1b) shows that the vertical displacement imposed to the wheel by a remarkable soil irregularity is converted into a smaller displacement of the rocker passive joint and therefore of the whole vehicle. The oscillations transmissibility turns consequently reduced (Fig. 2).

Fig. 1. a) Agri.q in a simulated rough terrain condition: the wheels system adopted allows a force distribution similar so that of a tracked vehicle; b) filter with respect to the oscillations.

290

P. Cavallone et al.

Fig. 2. Agri.q functional design

2.2 Landing Platform and Solar Panels Since the Agri.q is expected to interact with UAVs, it has been provided with an orientable platform, which allows drones landing over an always-horizontal surface even when the vehicle is moving on steep slopes. In fact, controlling the pitch angle αE and the roll angle αR both longitudinal and transverse ground slopes can be compensated, as shown in Fig. 3a). Since sustainability has been the leading concept in the development of the Agri.q, the landing platform is covered by solar panels, which ensure partial auto recharging of the batteries. Furthermore, the same positioning mechanism used for the platform orientation can be exploited to maximize the sunrays collection Fig. 3b).

Fig. 3. a) Landing platform; b) solar panels.

2.3 Robotic Arm According to the principles of precision agriculture, the Agri.q is expected to collect ground or leaves samples for lab analysis, maybe laying them down on the docked drone, and has to apply chemical treatments or fertilizers when and where it is necessary.

Design of the Mobile Robot Agri.q

291

Therefore, a lightweight 7 DOF collaborative robotic arm has been integrated into the rover. The base of the robotic arm is assembled on the back chassis, in order to increase its workspace. As a result, the end-effector can reach the ground to collect samples, see Fig. 4a) but also the drone docked on the landing platform. Furthermore, exploiting the mobility of the landing platform, it can reach leaves at heights up to 2.5 m above ground, as shown in Fig. 4b).

Fig. 4. a) The End effector of the robotic arm reaches the ground; b) end effector maximum height.

3 Design of the Actuation System The first application expected for the rover Agri.q is working in the vineyards. This kind of crops grows in a particular environment whose peculiarities lead the design requirements of the actuation system. A modular approach was adopted in order to simplify the design and reducing the cost. A chain system transmission has been chosen for its robustness and transmission capabilities. Moreover, it can operate in the dirtiest condition with a minimum maintenance. While going through the vineyards rows, the velocity of the rover must be limited to allow the accurate monitoring of the crops. The upper speed limit closely depends on the technology adopted for such task. In any case, the nominal speed of the vehicle has been set at 5 km/h. The rover must overcome rough terrains, in terms of both slope and unevenness. Only using the front driving units the rover has been designed to overcome slopes with inclination of αSoil = 15°. 3.1 Kinematic Analysis The transmission system of each driving unit is composed of an electric DC motor, a planetary gearbox and the aforementioned chain system. The angular speed required of the right motor ωM,R is defined by ωM ,R,Right = ωR,Right τ1 =

x˙ Right τ1 τ2 = ωW ,Right τ1 τ2 rW

(1)

292

P. Cavallone et al.

Fig. 5. Driving unit kinematics. Table 1. Parameters. Symbol

Description

Unite of measure

ωR

Output angular speed of the planetary gearbox

[rad/s]

τ1

Transmission ratio of the planetary gearbox

[/]

x˙ Right , x˙ Left

Driving unit speed

[m/s]

τ2

rP2 /rP1

[/]

ωW

Wheel angular speed

[rad/s]

The speed of the rover can be obtained from the follows equation (Table 1): x˙ =

x˙ Right + x˙ Left 2

(2)

3.2 Quasi-static Analysis Assuming a set of reasonable simplifications, a very simple dynamic model of the Agri.q can be drawn in order to address the motors choice. Figure 6 shows such a simplified approach. The rover is analysed in a quasi-static condition, while it is climbing a hill with a gradient of αsoil = 15° in order to define a maximum motor torque. Being the nominal speed of the rover limited at 5 km/h it is possible to neglect both the inertial effect and the rolling frictions among wheels and ground. With the hypothesis that the vehicle is symmetrical balanced left and the right side, the equilibrium along the x axis simply provides TF1 + TF2 = 1/2 Fp sinαSoil . The torque balance on the front wheel allows relating the driving torque CF to the soil reaction as: CF = TF rW (see Fig. 7a) and so CF1 + CF2 = FP /2sinαsoil rW . Considering the forces distribution of Fig. 7b, the driving torque can be estimated as CR = (CF1 + CF2 )rP1 /rP2 . Therefore, the required torque of the motor is CM ,R =

CR FP = sin αSoil rW τ1 2τ1 τ2

(3)

Design of the Mobile Robot Agri.q

293

Fig. 6. Half-rover static model.

Fig. 7. Free body diagram a) of the front wheel; b) force distribution among the three gears of a front driving unit.

3.3 Parametric Analysis The radius rP1 and rP2 depend on the number of teeth, respectively, Z1 and Z2 . On the one hand, Z1 should be as small as possible compatibly with the technological limits. Z1 = 16 was chosen. On the other, Z2 must be as high as possible, with the only upper limit of having a gear radius smaller than rW , to avoid interference issues with the soil. It was chosen Z2 = 48. The transmission ratio τ2 = rP2 /rP1 = Z2 /Z1 = 3 is defined. The radius of the wheel rW = 0,2 m has been defined employing a commercial mountain bike wheel. The size of the motor can be defined considering the power required PR of when the rover overcomes at the speed of 5 km/h = 1,4 m/s a climb with αSoil = 9° of inclination and a motor efficiency η = 0,9. Moreover, a more burdening yet brief using case was considered with a slope of α Soil = 15°. PM ,R =

ωM CM = 121 [W] η

A DC motor, with the following characteristics has been selected: PM ,Nom,1 = 120 W; PM ,Nom,2 = 160 [W] ωM ,Nom,1 = 315

rad ; ωM ,Max = 350 [rad/s] s

294

P. Cavallone et al.

CM ,Nom,1 = 0, 38 Nm; CM ,Nom,2 = 0, 51 [Nm] where the pedis 1 and 2 refer to the respective duty cycle ratings S1 (continuous duty) and S2 (short time duty, about 20 min). The motor gear choice has been then completed with a gearbox having a ratio τ1 = 15,88 in order to satisfy the requirements.

4 Preliminary Tests Preliminary tests have been executed to determine the actual operating conditions of the rover. A couple of test are shown herein: test 1 refers to reach a maximum speed on an even and flat path; test 2 evaluates the torques of the motor when the rover climbs a road with an inclination of αSoil = 9°. Both tests have been performed manually driving the rover. Therefore, the trajectories obtained are not perfect under the point of view of velocity or direction steadiness. Anyhow, some interesting results can be drawn as well. 4.1 Test 1 The Fig. 8 shows the angular speed of the right and left motor ωM,Right , ωM,Left , the current absorbed by the right and left motor IM,Right , IM,Left during the test 1.

Fig. 8. Angular speeds and absorbed currents of the front motors during test 1, flat path.

4.2 Test 2 Figure 9 shows the angular speed of the right and left motor ωM,Right , ωM,Left , the current absorbed by the right and left motor IM,Right , IM,Leftt during the test 2.

Design of the Mobile Robot Agri.q

295

Fig. 9. Angular speeds and absorbed currents of the front motors during test 2, uphill path.

4.3 Analysis of Results The speed of the rover can be determined from (1), (2) and from the Fig. 5 as it follows:   ω¯ M ,Right + ω¯ M ,Left rW π x˙ Right + x˙ Left = (4) x˙ = 2 2 τ1 τ2 30 The torque of the motor is related to the current I by the following equation: CM = kM I¯M

(5)

Where km = 0,055 Nm/A. During the test 1 (T1), the speed of the rover can be determined using (4)   km x˙ T 1 = 4, 7 h The test has been performed manually by a remote controller which is still in prototyping and the reference signal has been limited by a remote controller. During the test 2 (T2), the torque of the front motors can be calculated using the Eq. 5) CM ,Right,T 2 = 0, 52 Nm; CM ,Left,T 2 = 0, 45 [Nm] The torque of the motor can be compared with that coming from the simplified Eq. 2) with αSoil = 9°. CM = 0, 33 [Nm] Considering the efficiency of the motor and the chain of η = 0,9 and increasing of 2° the inclination due to the rolling friction α Soil = αSoil + 2°, (3) can be modified as it follows: FP  sinαSoil rW = 0, 44 [Nm] CM = η2τ1 τ1 The Fig. 8 and 9 show the difference between the angular speed of the right and the left motors. This fact causes a difference of torque and a difference of output power as P¯ M ,Right,T 2 = 86,4 W; P¯ M ,Left,T 2 = 73,5 [W]

296

P. Cavallone et al.

5 Conclusions Applications of precision agriculture are arousing the increasing interest of the robotics scientific community. The rover Agri.q has been expressly designed and prototyped to approach such tasks. In this paper, an overview of the robot mechanical designed is provided. The design guidelines adopted lead to the realization of a prototype which is nowadays in a mechanical test phase. Currently, different tests are still in program in order to validate the prototype in several ways. After such verification phase, the automatic drive and the control logics have to be defined and implemented. The paper also shows a couple of experimental tests, performed to validate the adopted design simplifications. The results show that the rover is able to fulfil the initial requirements (slope and velocity). Acknowledgements. We gratefully thank the PIC4SeR – PoliTO Interdepartmental Centre for Service Robotics.

References 1. ISPA. http://www.ispag.org. Accessed 06 Feb 2020 2. Wang, Y., Lan, Y., Zheng, Y., Lee, K., Cui, S., Lian, J.-A.: A UGV-based laser scanner system for measuring tree geometric characteristics. In: 5th International Symposium on Photoelectronic Detection and Imaging, Beijing, China (2013) 3. Zaman, S., Comba, L., Biglia, A., Aimonino, D.R., Barge, P., Gay, P.: Cost-effective visual odometry system for vehicle motion control in agricultural environments. Comput. Electron. Agric. 162, 82–94 (2019) 4. Khaliq, A., Comba, L., Biglia, A., Aimonino, D.R., Chiaberge, M., Gay, P.: Comparison of satellite and UAV-based multispectral imagery for vineyard variability assessment. In: Remote Sensing, vol. 11, no. 4, Switzerland (2019) 5. Chatzimichali, A.P., Georgilas, I.P., Tourassis, V.D.: Design of an advanced prototype robot for white asparagus harvesting. In: International Conference on Advanced Intelligent Mechatronics, Singapore, pp. 887–892 (2009) 6. Aljanobi, A.A., Al-Hamed, S.A., Al-Sushaibani, S.A.: A setup of mobile robotic unit for fruit harvesting. In: 19th International Workshop on Robotics in Alpe-Adria-Danube Region RAAD, Budapest, Hungary (2010) 7. Quaglia, G., Visconte C., Scimmi, L.S., Melchiorre, M., Cavallone, P., Pastorelli, S.: Design of the positioning mechanism of an unmanned ground vehicle for precision agriculture. In: Uhl, T. (ed.) Advances in Mechanism and Machine Science - Mechanism and Machine Science, vol. 73, pp. 2339–2348. Springer Nature Switzerland AG, Switzerland (2019) 8. Quaglia, G., Visconte C., Scimmi, L.S., Melchiorre, M., Cavallone, P., Pastorelli, S., Robot arm and control architecture integration on a UGV for precision agriculture. In: Uhl, T. (ed.) Advances in Mechanism and Machine Science - Mechanism and Machine Science, vol. 73, pp. 2339–2348. Springer Nature Switzerland AG, Switzerland (2019) 9. Quaglia, G., Cavallone, P., Visconte, C.: Agri_q: agriculture UGV for monitoring and drone landing. In: Gasparetto, A., Ceccarelli, M. (eds.) Mechanism Design for Robotics - Mechanism and Machine Science, vol. 66, pp. 413–423. Springer Nature Switzerland AG, Switzerland (2019)

Kinematic Design of an Adjustable Foot Motion Generator for Gait Rehabilitation Chanatip Thongsookmark1(B) , Agnes Beckermann2(B) , Mathias H¨ using2(B) , 1(B) and Yukio Takeda

2

1 Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan {thongsookmark.c.aa,takeda.y.aa}@m.titech.ac.jp Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen, Aachen, Germany {beckermann,huesing}@igmr.rwth-aachen.de

Abstract. Gait rehabilitation devices strive to adapt to the human body’s biomechanics and other characteristics. Not all patients have the same gait movement: it depends on their height, the length of their limbs, or other parameters. As a matter of fact, it has been shown that it is necessary for the rehabilitation device to be adjustable in both length and height separately in order to help the patient effectively. However, devices that are able to adapt to every patient’s body are usually very complex and/or expensive. This paper introduces new concepts for affordable foot motion generators that can separately and continuously adapt the length and the height of the generated gait trajectory. It especially focuses on a slider crank mechanism combined with a cam mechanism which allows for a compact, exact, and easily adjustable gait trajectory generation.

Keywords: Mechanism design

1

· Gait rehabilitation · Cam mechanism

Introduction

Research shows that strokes are one of the leading causes of disability, especially in the growing population of persons aged over 65 years [1]. After a stroke, most people encounter issues in walking and moving because the brain regions that coordinate these body movements are damaged [2]. In order for these people to recover their walking ability, rehabilitation is often required. In the last decades, rehabilitation technique has profited greatly from the new possibilities that robotics and greater computational capabilities have to offer. Gait rehabilitation robots allow patients to recover and walk again. They are able to perform repetitive movement with adequate accuracy, as well as reduce the cost of treatment and compensate a lack of therapists, therefore making rehabilitation more accessible [3]. However, even though commercial gait rehabilitation robots have shown significant therapeutic benefits, their cost is still a significant problem that makes c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 297–304, 2021. https://doi.org/10.1007/978-3-030-58380-4_36

298

C. Thongsookmark et al.

them inapplicable for rehabilitation clinics and home [4]. For small medical facilities, rehabilitation devices must come at a low initial and maintenance cost and must be able to adapt to any personal gait pattern for every patient. Devices are required to support the ankle joint while providing the appropriate movement with sufficient accuracy and consistency. Such a device should have a small footprint and be easy to use. Many researchers designed low cost gait rehabilitation devices with less sensors and actuators by applying link and cam mechanisms [3,5]. In [6], a gaitguidance device composed of a cam and Scotch-yoke mechanism that could be scaled to address the gait retraining needs of individual has been proposed. This device showed the possibility of scalable gait-like trajectory generator for personal gait rehabilitation. However, only one trajectory pattern could be generated and scaled. This does not allow to fully personalize the gait movement. The objective of this paper is to present the design of an affordable and adjustable foot motion generator for personal gait rehabilitation. The proposed mechanism is a compact mechanism composed of a slider crank mechanism with a cam mechanism, being able to easily scale the gait length and gait height separately.

2

Preliminary Mechanism Design

In order to increase the capability of personalization, the design focuses on the adjustment system of low-cost foot motion generator. As a matter of fact, the step length has to be able to vary between 500 mm and 800 mm to cover the step length of most people [7]. This can be observed in The GaitPhase Database [8], which offers measurement data of healthy ankle positions while treadmill walking. Due to the similarity of gait kinematics between children and adults, and similarity of normalized ankle path in gait trajectory [9,10], a sample ankle trajectory can be chosen and scaled up or down to match the ankle path of various gait trajectories. A sample sagittal-plane marker position data of the ankle path is shown in Fig. 1 (a). Figure 1 (b), (c), and (d) show samples of ankle path with scaling up in step length, step height, and step length and height, respectively. The adjustable foot motion generator is designed to work together with a treadmill and a body weight support system as shown in Fig. 2 (left). The upper body is carried by the body weight support system, so that the patient does not have to carry their body weight. The foot motion generator generates the ankle path and guides the user’s ankle along a desired target path. The device should be of a low complexity and should be able to be operated by a single actuator to keep the cost low. Another design requirement is a small footprint, as the device is designed for being used in small medical facilities or at home. We designed several devices following these design requirements. A slider crank - cam - slotted bar mechanism was chosen as the final design, as it was the smallest one among candidates, and the adjustment of both step length and width can be independently done with the same method.

Kinematic Design of an Adjustable Foot Motion Generator

299

Fig. 1. (a) A sample ankle position in gait trajectory (b,c,d) scaled up gait trajectories

Fig. 2. (left) Overview of the complete system and (right) a sketch of slider crankcam-slotted bar mechanism for foot motion generation

The slider crank - cam - slotted bar mechanism consists of two systems: the horizontal motion generation system and the vertical motion generation system, which are driven by the same input as show in Fig. 2 (right). The slider crank mechanism is used to generate a translational output motion in order to mimic

300

C. Thongsookmark et al.

the horizontal motion of human foot (called X-movement). The vertical motion is more complex, and is therefore realized by a cam mechanism, which generates a linear vertical output (Y-movement). Combining these mechanisms, the output motion in horizontal and vertical directions can be generated by one input and still be independently adjusted by two adjustment systems. Both adjustment systems consist of a slotted bar with an adjustable rotation center position system. By moving the center of rotation of the slotted bar, the output motion can be scaled up or down. The scaling ratio is the ratio between the lengths of the output and input sides of the slotted bar. The major disadvantage of this mechanism is that the slotted bar length needs to be long to keep appropriate force transmission angle.

3

Kinematic Design of the Slider Crank-Cam-Slotted Bar Mechanism

The dimensions of the mechanism are given by the trajectory of the end effector, as well as by the space limitations. This section will describe how the dimensions of the mechanism can be determined. To simplify the initial synthesis process, the ratio between output and input sides of both slotted bar is set to 1, and minimal step length of 500 mm is considered. The slider crank can be described with three parameters: the length of the crank l1 , the length of the connecting rod l2 and the excentricity of the crank e. To simplify the next steps, the excentricity will be set to zero. Therefore, the = length of the crank l1 is directly in relation to the step length: l1 = steplength 2 250 mm. The smallest connecting rod is l2 = l1 / sin(90◦ − μcrank ), where μcrank is the minimal transmission angle authorized for a slider crank mechanism. In order to determine the cam dimensions, the position of the follower has to be known for every possible input ψ. Considering the X-transmission ratio of 1, the X-motion of the end effector X(ψ) can be expressed according to Eq. (1), with X0 being the initial position of the end effector.  X(ψ) = X0 − (l1 · cos(ψ) + l2 2 − (l1 · (ψ))2 ) (1) The end effector follows the cyclic gait trajectory introduced in Sect. 2 where for every X-Position of the end effector, there are two possible Y-Positions, except in the turning points X(0) = X0 and X(π) = X0 + l1 + l2 . One of these positions corresponds to the upstroke of the slider crank, the other one to the downstroke. As ψ is given, the corresponding Y-Position Y (ψ) can be calculated numerically by interpolating the trajectory. As the Y-Transmission ratio is set to 1, the position of the roller follower yr (ψ) = yr (0) − (Y (ψ) − Y (0)) results directly from Y (ψ). The initial position of the roller follower depends on the missing dimensions of the mechanism such as the excentricity of the follower, as well as the base radius of the cam. The base radius is chosen in a way that the transmission angle stays over 50◦ . Once the position of the roller follower yr (ψ) is known for a number nψ of different cam orientations, the cam curve can be determined. Multiple methods

Kinematic Design of an Adjustable Foot Motion Generator

301

exist for this step [11,12]. At the IGMR, a complex method described in the VDI Guideline [13] has been implemented with success in different projects [14]. This method also allows to directly have access to the transmission angle. However, in order to be able to be implemented efficiently, all methods require the movement of the roller follower to be not only continuous, but also have its velocity and acceleration be continuous. First results show that the movement generated by the gait data is not sufficiently continuous around ψ = 0 and ψ = 2π. As a matter of fact, yr (ψ) is continuous, but it’s first derivative is not. This can be explained by the fact that the trajectory introduced in Sect. 2 is based on measurements. The movement is not perfectly periodic. Therefore, the roller follower movement has to be smoothed before being used to calculate the cam profile. This introduces a small positioning error, that will have to be considered when interpreting the results.

4

Results

Fig. 3. Representation of a possible mechanism, built according to Sect. 3

Figure 3 shows the structure of a mechanism designed according to the steps from Sect. 3. Because of the way it was designed, all the initial conditions are respected. The adjustment mechanism allows for a variation of the step length and step width, the transmission angles stay over their respective minimal values, the mechanism can be limited to a certain size, and the cam parameters allow for a manufacturable cam. However, due to the smoothing of the roller follower movement, the generated end effector trajectory varies slightly from the original measured gait trajectory. As these two trajectories are not necessarily defined by the same amount of points, or at the same interval, an appropriate measuring tool has to be chosen. A possible metric to compare the two trajectories is to compare the two enclosed areas. Another metric to describe the relative distance between two sets of points is the Hausdorff distance, defined in Eq. (2).   (2) dH (A, B) = max sup inf d(a, b), sup inf d(a, b) a∈A b∈B

b∈B a∈A

Here, A is the set of points of the generated end effector trajectory and B the original measured set of points. The distance is the maximal distance from any

302

C. Thongsookmark et al.

point of one of the two sets to the polygon of the other set. Therefore, dH (A, B) shows the maximal deviation between the calculated and the goal trajectories. By analyzing the trajectory data from the mechanism presented in Fig. 3, we find the area error to be of 218 mm2 , which represents a relative error of 0.56%. The Hausdorff distance is 4.092 mm and if expressed relatively to the step length represents also an error of less than 1%. Both errors are within an acceptable range, even though a smoothing factor of 10 was used. This means that the necessary smoothing of the roller follower movement doesn’t induce a very high error. If for one reason or another the error needs to be minimized, the smoothing function can be picked differently. It is also important to notice that the transmission ratio of both movements can easily be changed by changing the position of the rotation center of the adjustment joints. Fig. 4 shows how the same mechanism is able to generate a gait movement with a step length of 800 mm by moving the center point of the X-Adjustment slotted bar closer to the slider.

Fig. 4. Adjustment of the step length (here 800 mm)

5

Optimization Possibilities

The previous sections describes the kinematic design process in the case where the excentricity of the crank is set to zero. However, changing the dimensions of the crank slider mechanism directly influences the X-motion of the end effector X(ψ). For instance by adding an excentricity, the upstroke can be made shorter or longer. This allows us to influence the motion of the roller follower, and therefore also the transmission angle of the cam mechanism. This influence can not easily be quantified. It can however be used to maximize the minimal transmission angle. It has been shown that varying the drive of a mechanism (for instance with the help of servo drives [15]) can be used to optimize the mechanism. Here, we can do the same by varying the drive function X(ψ). By implementing such an optimization in Matlab, it is possible to show that there are other combinations of the mechanism that allow for a much higher cam transmission angle. For instance, introducing a crank excentricity e = −500 mm and setting the lengths to l1 = 135.6 mm and l2 = 645.6 mm allows us to significantly augment the minimal transmission angle (see Fig. 5). However, a

Kinematic Design of an Adjustable Foot Motion Generator

303

mechanism with such a great excentricity would necessitate a much larger installation space. Moreover, the minimal transmission angle of the slider crank might become critical as well. By limiting the excentricity to 100 mm, the minimal transmission angle can still be maximized, but gives results that are very close to the case where e = 0 mm (see Fig. 5).

Fig. 5. Cam transmission angles for three different crank excentricities

The dimensions of the slider crank are not the only parameters that can be considered when optimizing the mechanism. Changing the transmission ratios would also influence the forces in the mechanism and might lead to another optimal configuration. In order to exactly understand which transmission angle might become more critical, a full kinetostatic analysis would be necessary. It would allow to calculate the actuation torque for the motor and to optimize this value, as it is much more significant for the mechanism.

6

Conclusion

The research done in this paper was able to highlight the need for a new kind of gait rehabilitation device due to the rising demand for cheap, simple, yet adaptive devices. The use of cam mechanisms has proven to be one of the best possible solutions for this new problematic. In the primary design process, a novel mechanism with an adjustment system for the step length and height was presented. This mechanism consists of a combination of a slider crank and a cam mechanism. The paper sums up the kinematic design process and presents results of the kinematic analysis. These results show that the mechanism is able to satisfy the demand for a reliable, adaptive mechanism: it is able to realize all the wanted trajectories. Although these first results are satisfying, further analysis has shown that a more thorough optimization of the mechanism could lead to a more efficient and compact build. Such an optimization should also consider the internal forces and torques, which still need to be determined.

References 1. He, Q., Wu, C., Luo, H., Wang, Z., Ma, X., Zhao, Y., Lu, J., Xiang, C., Qin, Y., Wu, S., Yu, F., He, J.: Trends in in-hospital mortality among patients with stroke in China. PLoS One 9, e92763 (2014)

304

C. Thongsookmark et al.

2. Lee, H., Jung, S.: Prediction of post-stroke falls by quantitative assessment of balance. Ann. Rehabil. Med. 41(3), 339–346 (2017) 3. Shao, Y., Xiang, Z., Liu, H., Li, L.: Conceptual design and dimensional synthesis of cam-linkage mechanisms for gait rehabilitation. Mech. Mach. Theory 104, 31–42 (2016) 4. Shin, S.Y., Deshpande, A.D., Sulzer, J.: Design of a single degree-of-freedom, adaptable electromechanical gait trainer for people with neurological injury. J. Mech. Robot. 10, 044503 (2018). https://doi.org/10.1115/1.4039973 5. Alves, P., Cruz, F., Silva, L., Flores, P.: Synthesis of a mechanism for human gait rehabilitation: an introductory approach. Mech. Mach. Sci. 24, 121–128 (2015) 6. Nelson, C., Stolle, C., Burnfield, J., Buster, T.: Synthesis of a rehabilitation mechanism replicating normal gait. In: 14th IFToMM World Congress, Taiwan, pp. 25–30 (2015) 7. Gill, S., Keimig, S., Kelty-Stephen, D., Hung, Y., DeSilva, J.: The relationship between foot arch measurements and walking parameters in children. BMC Pediatr. 16, 15 (2016) 8. GaitPhase Database. https://www.mad.tf.fau.de/research/activitynet/gaitphasedatabase/. Accessed 12 Dec 2019 9. Ganley, K., Powers, C.: Gait kinematics and kinetics of 7-year-old children: a comparison to adults using age-specific anthropometric data. Gait Posture 21(2), 141– 145 (2005) 10. Oberg, T., Karsznia, A., Oberg, K.: Joint angle parameters in gait: reference data for normal subjects, 10–79 years of age. J. Rehabil. Res. Dev. 31(3), 199–213 (1994) 11. Flores P.: A computational approach for cam size optimization of disc cam-follower mechanisms with translating roller followers. J. Mech. Robot. 5 (2013). https:// doi.org/10.1115/1.4025026 12. Rothbart, H.A.: Cam Design Handbook. McGraw-Hill, New York (2004) 13. VDI 2142-2, Auslegung ebener Kurvengetriebe Blatt 2, Standard, Verein Deutscher Ingenieure, Beuth, Berlin (2011) 14. Mueller M., Huesing M., Beckermann A., Corves B.: Linkage and cam design with MechDev based on non-uniform rational B-Splines. Machines 8, 5 (2020). https:// doi.org/10.3390/machines8010005 15. Mueller, M., Hoffmann, M., Huesing, M., Corves, B.: Using servo-drives to optimize the transmission angle of cam mechanisms. Mech. Mach. Theory 135, 165–175 (2019). https://doi.org/10.1016/j.mechmachtheory.2019.02.001

Design and Construction of the DragonBall Bir Bikram Dey(B) and Michael Jenkin(B) EECS, Lassonde School of Engineering, York University, Toronto, ON, Canada {bikram,jenkin}@eecs.yorku.ca

Abstract. Spherical robots provide a number of advantages over their wheeled counterparts, but they also presents a number of challenges and complexities. Chief among these are issues related to locomotive strategies and sensor placement and processing given the rolling nature of the device. Here we describe DragonBall, a visually tele-operated spherical robot. The DragonBall utilizes a combination of a geared wheel to move the center of mass of the vehicle coupled with a torque wheel to change direction. Wide angled cameras mounted on the robot’s horizontal axis provide a 360◦ view of the space around the robot and are used to simulate a traditional pan tilt zoom camera mounted on the vehicle for visual tele-operation. The resulting vehicle is well suited for deployment in contaminated environments for which vehicle remediation is a key operational requirement. Keywords: Spherical robots

1

· Robot control

Introduction

Unmanned vehicles are popular for autonomous operation, especially in hazardous environments. Unfortunately robot deployment in such environments can be challenging. Vehicles can be difficult to deploy and upon mission completion may require decontamination. A key issue in terms of remediation is that penetration of the vehicle body itself by the drive mechanism and sensors provides potential risks for environmental contaminants to enter into the vehicle. Sealing shafts, sensors and the like from external environment contamination can be difficult for many “off the shelf” autonomous robots. One vehicle design that avoids many of the issues associated with environmental contamination is the spherical robot design. Here the entire robot is enclosed, it has a “ball-like” shape and is sealed from the outside environment. As a consequence cleaning and decontamination after deployment can be limited to the outside surfaces of the vehicle. Also due to its shape, the robot can be easily deployed in complex spaces and environments that are challenging for some wheeled or tracked vehicles. For example, a spherical robot can be deployed to go into pipes and narrow corners easily, which can be a critical requirement. Such maneuverability enables Supported by NSERC and the National Canadian Robotics Network. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 305–312, 2021. https://doi.org/10.1007/978-3-030-58380-4_37

306

B. B. Dey and M. Jenkin

Fig. 1. Spherical robot classes.

a spherical robot to perform tasks that may not be suitable for its wheeled or tracked alternatives. Another key advantage of a spherical robot design is that a spherical robot “cannot be overturned” [2]. This is a great advantage for autonomous robots operating in uncertain and complex environments and can simplify vehicle deployment. Although they have many positive characteristics, spherical robots present a number of interesting challenges. Their relative mechanical complexity has limited their development and their continuously rolling surface presents a challenge for sensor deployment on the vehicle. As a consequence common autonomous vehicle tasks such as visual tele-operation are in general poorly understood for such vehicles. Here we describe DragonBall a visually tele-operated autonomous robot. DragonBall exploits a combination of strategies for locomotion and a pair of ultra wide field of view cameras to simulate a standard pan, tilt and zoom camera for visual tele-operation.

2

Previous Work

Spherical robots can be divided in two main design strategies, active and passive designs, as done by Tomik [11] and as shown in Fig. 1. Passive design robots rely on some external force to move them. Examples of passive spherical robots include those which are powered by an external source such as the ones powered by wind. See Thistle [10] and Mars Ball [12] for example. In contrast, active design vehicles use an active mechanism to change state. Active designs can be further divided into deformable and rigid shell designs. Spherical robots which can morph to different shapes to move forward, such as the robot developed by Sugiyama [9] are categorized as deformable spherical robots. This robot class deforms from the spherical shape in order to move. Perhaps the most common spherical robot design relies on a rigid shell. Various rigid shell vehicles have been built with different types of internal drive systems. Most rigid shell spherical robots rely on either conservation of angular momentum, mass unbalance or direct drive to drive the robot. The drive system, or “Inside Drive Unit” (IDU) [2] of a spherical robot can be divided in three main classes, as shown in the bottom layer of the classification chart given in Fig. 1.

Design and Construction of the DragonBall

(a) Dragon Ball

307

(b) Its internal structure

Fig. 2. The DragonBall and its internal structure

Numerous studies have been performed to evaluate the viability of spherical robots (see [4–6] for example). Quite often such research platforms are designed to work only within the lab or in a similarly structured indoor environment. The DragonBall however, was built to validate the design of a spherical robot platform for contaminated indoor and outdoor environments. Unlike a typical indoor robot it is important for an outdoor robot to be robust to a wide range of conditions. This introduces further design challenges. For example mobile robots operating in an outdoor experiment can be damaged due to mishandling, bad weather, rough terrain, etc. To take one aspect of this challenge, consider the structure of the spherical shell. Many existing spherical robots are built with an outer shell constructed from acrylic. The acrylic dome provides the robot with two things; a structural shape and a transparent exterior. While acrylic offers a clear view of the outside, it can easily lose its transparency by becoming scratched or marked from rough surfaces. It is also not a particularly strong material, a concern when deployed in complex outdoor environments.

3

Physical Design and Construction

The basic requirement behind designing the DragonBall was to design and build a robot that can carry sensors within a sealed spherical shell. Since transparent spheres are not an ideal solution, having transparent sensor ports on the sphere so that cameras can view the environment is critical. This leads to a shell design similar to that of the Rotundus (Groundbot) vehicle [7]. Such ports also provide access for wireless communication and a port for charging and wired access to internal electronics. The resulting design of DragonBall is shown in Fig. 2 and Fig. 3.

308

B. B. Dey and M. Jenkin

Fig. 3. Overall assembly of DragonBall.

3.1

Shell

The external shell provides the rolling contact surface and protects internal components. Composite building materials such as fiberglass and carbon fiber, are popular construction materials for smoothly changing surfaces such as the robot’s shell. For the DragonBall carbon fiber was chosen over fiberglass as it provides more stiffness. This choice offers increased strength and better scratch resistance while providing similar weight as fiberglass. Although the DragonBall robot is light in weight, having a strong body ensures that if operations such as falling from a height are required, the robot will be able to meet this requirement. Operation of this robot will not only be limited to land. The vehicle should also be able to navigate on the surface of bodies of water. If the design is watertight, travelling over water will be possible. Modifying this robot for operation in contaminated environments will also be straightforward as the robot has no penetration of the shell. Amongst the various approaches of manufacturing carbon fiber parts there are three different methods which are commonly used [1]. The first one is called hand layup or wet layup. This is the most simple and widely used method. The second method is resin infusion layup, where a vacuum is used to press the carbon fiber sheet to a mold while drying. In oven cured prepeg layup, sheets of carbon fiber cloth with pre-impregnated resin on their back are used. For the DragonBall the hand-layout method was used. Carbon fiber cloths come in various weights and fiber layouts. The strength of the final design depends on the orientation of the fiber to the load path [1]. The ability of a fabric to conform to a shape is also know as its drapability. As plain weaves “are not very drapeable” [1] compared to twill weave, for this project the twill weave was chosen. Bi-directional carbon fiber was used to handle multidirectional loads [1]. 3.2

Drive System

Given that there exists a large design space for spherical robots, it can be difficult to make informed design choices without prototyping a full system and using

Design and Construction of the DragonBall

309

this prototype to identify strengths and weaknesses of different design choices. Recognizing this, a prototype robot was built first, which provided insights into designing and manufacturing a spherical robot. It also helped in understanding the manufacturing capability given the resources available. The prototype was called “Spherico” (full details of Spherico appear in [3]). Spherico provided an essential test bed for the design of the DragonBall robot. Spherico used a track drive system to manipulate the centre of mass of the vehicle. Although this provided effective, it was found that sharp jolts of the drive mechanism could damage the drive wheels relative to the track. So for DragonBall a track drive system was adopted but a more effective (and rigid) track support was used and a shock absorption system was adopted to reduce torque from the flywheel system from impacting the drive system (see Fig. 2(b)). 3.3

Turning

Several spherical robots have been built using systems that use conservation of angular momentum to change the direction of the vehicle. The DragonBall also uses this approach. Commonly used turning systems for spherical robots are the reaction wheel, momentum wheel and control moment gyroscope (CMG) [8]. The reaction wheel uses the reaction torque from a flywheel when it is accelerated to turn the vehicle. The reaction wheel is normally not rotating and only rotates when the vehicle itself is rotating. This requires a motor with high torque. In the momentum wheel system uses a motor to continuously rotate the flywheel at high speed. A braking system is then used to stop the flywheel quickly to exchange momentum [8]. While an additional braking system is required in this system, the motor does not need to have high torque, hence a smaller motor can be used. Control moment gyroscope (CMG) is widely used in satellites and spacecraft. Unlike the previous two systems in a CMG system the flywheel’s speed remains constant and instead precession torque generated by the flywheel is harnessed. Having a flywheel inside the robot means that there will be torque transfer between the IDU and the shell. A system is required to withstand this torque. Four rollers were installed on each side of the drive track to safely transfer the force. 3.4

Electronics

Although the current version of the DragonBall operates in a tele-operated fashion, in the long term it is intended to operate the vehicle autonomously. There exist a number of small footprint computer systems that could be used to provide on-board computing. The DragonBall uses the NVIDIA TX2. The TX2 only performs the minimum amount of low level tasks with most low level task distributed to dedicated micro-controllers.

310

B. B. Dey and M. Jenkin

(a) Reaction system

(b) CMG system

Fig. 4. Time lapse images of different turning strategies with the DragonBall

The DragonBall incorporates a number of dedicated sensor systems including: – Cameras- The DragonBall uses two Pixpro 4K to collect data of the surrounding environment. – RTK GPS- GPS positioning is used to accurately track the robot’s motion – Odometer- The odometer in conventional wheeled vehicles is integrated into the wheel. For a spherical robot odometer sensor needs to be installed between the IDU and the external shell. – Torque sensor- The controller driving the flywheel, requires the speed profile of the flywheel more than the position of the flywheel, in order to be used as a system for angular momentum transfer. – IMU- The DragonBall couples a IMU together with a temperature sensor, compass and an ARM Cortex-M0 based processor that is used for performing sensor fusion to estimate vehicle state. – Current and Voltage- The DragonBall has two battery sets. Sensing battery voltage is critical not only for the operation of the robot but also due to the fact that a LiPo battery can be damaged permanently if it is operated under certain a voltage. On the DragonBall each battery set has its own system of power monitoring.

4

Visual Tele-Operation

Torque for driving the robot forward comes from the moment created by the mass moving up the track [3]. If the mass maintains a constant angle from the gravity vector then the shell will get a constant force from the IDU. Keeping a constant angle is a complex task for a human operator, so direct vehicle control by adjusting the position of the track motor is impractical. Instead, for vehicle tele-operation the set point is defined in terms of the orientation offset of the drive system motor. Turning the vehicle is controlled by breaking or accelerating the flywheel.

Design and Construction of the DragonBall

311

Fig. 5. Robot’s view of the outside world. On the left, images from DragonBall cameras have been dewarped, blended at the seam and stitched together. Left side shows zoomed in view of the selected area using the PTZ system.

Due to the distribution of the mass around the track, the body of the robot balances itself at +10◦ on a horizontal plane. To drive the vehicle at a constant acceleration the center of gravity is maintained at a constant orientation offset. Rolling resistance changes from surface to surface. Unlike in a lab environment where spherical robots are typically used on a floor, running a spherical robot in the outdoor environment requires accounting for rolling resistance as well. When trying to drive the robot at a low speed from a full stop, the robot may not roll forward due to rolling friction. This lack of motion can be detected from odometry data. If a situation like this happens, the robot oscillates the set point of the drive motor backwards and then forwards to the desired angle in order to produce a vehicle acceleration that overcomes the rolling resistance and the vehicle starts to move. Turning the vehicle when it is stopped is straightforward. Braking the flywheel causes the vehicle to turn in one direction while accelerating it causes it to turn in the other. This produces the smallest turning radius. Although this two system requires the robot to stop and then turn. Another method is to accelerate while the flywheel is turning, thus causing the robot to tilt in one side (due to precession) which causes the vehicle to turn but with a larger turning radius. Figure 4 shows sample turning trajectories. The two wide field of view video cameras produce an overlapping field of view that completely surrounds the vehicle from approximately 1 m from the vehicle’s centre. The two camera images can be merged to form a panoramic view around the vehicle and this panorama can be sampled appropriately to provide a simulated standard camera view from the centre of the robot in any direction and with a software zoom function. Figure 5 shows the panorama and the simulated PTZ camera view from the robot.

312

B. B. Dey and M. Jenkin

During visual tele-operation of the vehicle the operator is provided with a vehicle centric control of the simulated PTZ camera obtained from the two ultra wide field of view cameras. A second joystick provides acceleration and rotational control of the vehicle.

5

Summary and Future Work

DragonBall is a fully enclosed spherical robot with enclosed actuation and sensor systems. In addition to IMU and GPS sensors, the robot also relies on two wide field of view cameras mounted in ports along the vehicle’s axis. These cameras provide a panoramic view of the space around the robot and can be used to provide a simulated PTZ camera. This simulated PTZ camera coupled with a control system that allows a driver to choose a vehicle acceleration and turning rate is an effective control from vehicle tele-operation. Ongoing work with DragonBall is exploring SLAM with wide field of view cameras and IMU/GPS sensors.

References 1. Barbero, E.J.: Introduction to Composite Materials Design. CRC Press, Boca Raton (2017) 2. Crossley, V.A.: A literature review on the design of spherical rolling robots, Pittsburgh, PA, pp. 1–6 (2006) 3. Dey, B.B., Jenkin, M.: Spherico: rapid prototyping a spherical robot. In: IEEE International Conference on Information and Automation, Wuyishan, China (2018) 4. Guo, S., Mao, S., Shi, L., Li, M.: Development of an amphibious mother spherical robot used as the carrier for underwater microrobots. In: Complex Medical Engineering (CME), ICME International Conference, pp. 758–762. IEEE (2012) 5. He, Y., Guo, S., Shi, L., Pan, S., Wang, Z.: 3D printing technology-based an amphibious spherical robot. In: IEEE International Conference on Mechatronics and Automation, pp. 1382–1387 (2014) 6. Hern´ andez, J.D., Barrientos, J., del Cerro, J., Barrientos, A., Sanz, D.: Moisture measurement in crops using spherical robots. Ind. Robot: Int. J. 40(1), 59–66 (2013) 7. Kaznov, V., Seeman, M.: Outdoor navigation with a spherical amphibious robot. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (2010) 8. Schroll, G.C.: Dynamic model of a spherical robot from first principles. Ph.D. thesis, Colorado State University. Libraries (2010) 9. Sugiyama, Y., Shiotsu, A., Yamanaka, M., Hirai, S.: Circular/spherical robots for crawling and jumping. In: Proceedings of the IEEE International Conference on Robotics and Automation. IEEE (2005) 10. Suomela, J., Ylikorpi, T.: Ball-shaped robots: an historical overview and recent developments at TKK. In: Field and Service Robotics, pp. 343–354. Springer (2006) 11. Tomik, F., Nudehi, S., Flynn, L.L., Mukherjee, R.: Design, fabrication and control of spherobot: a spherical mobile robot. J. Intell. Robot. Syst. 67(2), 117–131 (2012) 12. Ylikorpi, T.: A biologically inspired rolling robot for planetary surface exploration, vol. 70. Degree of Licentiate of Technology, Helsinki University of Technology, Department of Automation and Systems Technology, Automation Technology Laboratory (2005)

Wire-Pulling Mechanism with Embedded Soft Tubes for Robot Tongue Nobutsuna Endo(B) Tokyo Denki University, Tokyo, Japan [email protected]

Abstract. There do not exist enough physical models of human oral and laryngeal systems for human speech movement in both computer simulators and mechanical simulators. In particular, there is no robot tongue mechanism that completely reproduces the deformation motion of the human tongue. The human tongue has no skeleton inside and is an aggregate of tongue muscles. A driving mechanism that can drive and control the deformation of a soft body like the human tongue with multiple degrees-of-freedom and has an affinity for the soft body has not been realized. In order to solve this problem, this paper proposes a wirepulling mechanism with embedded soft tubes. By using this, it is possible to realize a flexible tongue mechanism that can be deformed with multiple degrees-of-freedom without the wire breaking the flexible tongue. A two degrees-of-freedom input planar mechanism capable of contraction and bending was prototyped. The kinematic model under an assumption of piecewise constant curvature was formulated. An experiment confirmed that the durability was improved compared to the one without the soft tubes. A deformation test confirmed that the prototype was contracted and bent like the model. However, the tip angle error was not small enough. More accurate modeling and deformation sensors are the future work. Keywords: Tongue

1

· Vocal robot · Wire-driven mechanism

Introduction

Motor functions in the human oral and laryngeal systems include speech, mastication, swallowing, and sucking. Analyzing the movements of the oral system (pharynx, upper jaw, palate, tongue, lower jaw, teeth, and lips) and the laryngeal system (around the vocal folds and epiglottis) is more difficult than the motion analysis of the limbs, trunk, and head, because most of these organs exist inside the body, so that it is impossible to visually observe these area. Although there are observation methods using MRI and X-ray contrast, they can only measure This work was partially supported by JSPS KAKENHI Grant-inAid for Early-Career Scientists 18K13717 and Research Institute for Science and Technology of Tokyo Denki University Grant Number QT17T-04. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 313–320, 2021. https://doi.org/10.1007/978-3-030-58380-4_38

314

N. Endo

the structure in a stationary state or follow a global change in position and orientation in a short time, and cannot observe the movement state for a long time and the force applied to each part. There are approaches that can be reproduced by computer simulation, but due to the non-linearity of the mechanical properties of each organ in the living body, it has not been possible to construct a sufficient comprehensive simulation model. There also exists an approach of reproducing as a physical model in the real world by robotics: Waseda Talker robots [1], a vocal robot [2], a mastication robot [3]. In particular, the Waseda Talker series was limited to the speech function, succeeded in uttering vowels and some consonants due to vocal cord vibration and deformation of the vocal tract shape in the oral cavity, and is the best reproduction of human’s oral and laryngeal systems. The tongue mechanism of the Waseda Talker robot is based on static MRI measurement of the shape of the tongue when uttering Japanese five vowels. The drive mechanism consists of a slider crank mechanism and a skin covering the tongue surface made of a soft material. Therefore, it is not suitable for reproducing tongue shapes and movements with large curvature during consonant utterance. The human tongue is an aggregate of muscles without a skeleton inside, except a small hyoid. The human tongue is capable of complex and flexible deformation movements by the mechanism in which some tongue muscles contract and the other tongue muscles elongate. In order to reproduce the flexible structure and deformation motion of the human tongue, the author considered that a wire-pulling mechanism that drives the flexible tongue is suitable. The only previous study on tongue deformation driven by pulling wire is the utterance robot Anton [4]. In this robot, the tongue is made of soft silicone resin in order to reproduce the softness of human tongue, and the motion of the superior longitudinal tongue muscle, one of the internal tongue muscles, is reproduced with a single wire. Simply embedding the wire in the soft silicone resin will cause that the wire tears the resin. At Anton, by attaching several beads to the wire, the wire does not tear the resin and converts the wire-pulling motion to the tongue deformation motion. However, Anton’s tongue has only one degree-of-freedom (DOF) because it is difficult to embed many wires with beads at the time of tongue molding. On the other hand, the wire-pulling mechanism used in an anthropomorphic soft hand by the author [5] embeds hard tubes in the soft silicone resin so that the wire does not tear the soft material. Although, this mechanism can realize a localized deformation like a knuckle, and cannot realize spatially continuous deformation like a tongue. This paper proposes a wire-pulling mechanism with embedded soft tubes (Fig. 1) as a method to incorporate wires into soft material. Due to this method, a flexible robotic tongue, capable of deforming in multiple degrees-of-freedom, can be realized without the wire tearing the tongue. Also, since the soft tube can be fixed and embedded during molding, it is easier to manufacture than the Anton method. This paper reports its structure, the kinematic modeling, the durability test, and the deformation test.

Wire-Pulling Mechanism with Embedded Soft Tubes for Robot Tongue

(a) Structure

315

(b) Movement

Fig. 1. Basic idea of the structure and the movement of 2-DOFs planar mechanism version of the proposed mechanism.

2

Mechanical Configuration

As a mechanism to deform a soft body without the wire tearing the soft body, this paper proposes a mechanism that embeds soft tubes that is harder than the base material of the soft body but can elongate, contract, and bend flexibly. Figure 1 illustrates the basic idea of the structure and the movement of 2-DOFs input planar mechanism. This mechanism is capable of the contraction movement and the bending movement by the combination of two wire tensions. Generally, the soft material that constitutes the main deformable part is worn out by the movement of the driving wires and is easily torn. In order to suppress the wear, a soft tube formed of a harder but deformable material is placed between the deformable part and the wire. This soft tube will also eventually wear out due to the wire, but it is considered to be more durable than the soft material composing the main deformable part. To verify the feasibility of this basic idea, a prototype shown in Fig. 2 was built. The main deformable part was made of a silicone resin (Dragon Skin FXPro. Smooth-On, Inc.) with a Silicone Thinner. The thinner compounding ratio to the resin was 50 wt% [6]. I used a silicone tube (Hardness: Shore A60. Outside diameter: 2 mm. Inside diameter: 1 mm. MGJG-1×2. MonotaRO Co., Ltd.) as the soft tube. The outside diameter of the wire was 0.45 mm (EA628SS-0.4. ESCO Co., Ltd.). The base and the cap were formed with PLA resin using a FDM 3D printer. The wires were pulled by position-controlled motors (RE35. maxon motor), harmonic drives (CSF-14-80-2XH-F. Harmonic Drive Systems Inc.), and pulleys (Diameter: 45 mm). The base and the pulleys were connected by Bowden cable mechanisms using a flexible tube made of PTFE (Outside diameter: 3 mm. Inside diameter: 1 mm. TUF-100. Chukoh Chemical Industries, Ltd.) as a Bowden sheath. Figure 3 shows the deformation.

3

Kinematic Modeling

A deformation model under the assumption of a piecewise constant curvature (PCC) [7] is derived with reference to [8]. Figure 4 shows the deformation model

316

N. Endo

(a) Appearance

(b) Structure

(c) Main dimension

Fig. 2. Prototype of the 2-DOFs input planar mechanism.

(a) Natural state

(b) 50 % contraction

(c) 90 deg bending

Fig. 3. Contraction and bending of the prototype.

of the deformable part (50 × 24 × 20 mm) in Fig. 2. Let d be the distance between the wires, and L be the length of the deformable part when the displacements of the wires are zero. Let the lengths of the wires in the deformable part be l1 and l2 , respectively, and the actuator space be q = [l1 l2 ]T . According to the PCC assumption, the neutral line l, the wire l1 , and the wire l2 , form arcs centered at point (0, ρ). The following equation holds for the radius of curvature ρ using the cap orientation θ, ⎧ ⎪ ⎨ l = ρθ l1 = (ρ − d2 )θ (1) ⎪ ⎩ d l2 = (ρ + 2 )θ Solving this yields a transformation from the actuator space to the configuration space c = [l θ]T .    1 1  l(q) l1 c(q) = = 21 12 (2) θ(q) − d d l2 Likewise, the actuator space can be transformed from the configuration space by the following equation.      1 − d2 l l (c) = (3) q(c) = 1 l2 (c) θ 1 d2

Wire-Pulling Mechanism with Embedded Soft Tubes for Robot Tongue

[x

317

The transformation from the configuration space to the task space p = y]T is as follows. ⎧ l ⎪ ⎨ x = ρ sin θ = sin θ θ ⎪ ⎩ y = ρ(1 − cos θ) = l (1 − cos θ) θ

(4)

Fig. 4. Deformation model of the 2-DOFs input planar mechanism of the proposed mechanism. The light pink part shows the natural state, and the dark pink part shows the deformation state when the wires are pulled.

4

Durability Test

An experiment was performed to confirm whether the proposed soft tube embedding method has improved durability compared to the one without the soft tube embedded. One wire length l2 = L = 50 mm was constant, and the other wire l1 was pulled by the motor in a sine wave shape with a frequency 2 Hz so that the pulley angle was 0◦ to 90◦ . If the prototype is deformed according to the model above, the maximum wire length displacement of l1 is Δl1 = L − l1 = 35.3 mm, the length of the neutral line l is 32.3 mm, and the cap angle is θ = 1.77 rad = 101◦ . The average pulling speed is 141 mm/s and the maximum pulling speed is 222 mm/s. The number of cycles until a visible crack was generated on the outer surface of the wire l1 was measured. For one test subject, the experiment was performed at l1 , and then replaced with l1 and l2 , and a total of two trials were performed. The result is shown in the Table 1. Without soft tubes, that is, when the wire rubbed against the soft body directly, it took less than 200 times for the wire to tear at a minimum thickness of 1.5 mm from the inner wall of the wire’s path to the outer surface. On the other hand, it can be seen that the number of cycles increased more than 10 times when soft tubes are embedded. From the result, it can be said that the proposed method contributes to the improvement of the durability. However, in this test method, it was only

318

N. Endo

visually observed that the crack had occurred on the outer surface, and it was not known that the crack had occurred on the inner wall of soft tube before the outer surface of the soft body was torn. Even with the proposed method, it is about 2,000 times before cracks occur on the outer surface, which means that only about 1,000 times can be practically used under these conditions. This durability depends on the friction between the soft tube and the twisted wire. To further improve durability, the following methods can be considered: use more friction-resistant soft tubes, use non-twisted wires, and/or lubricate the inside of the soft tubes or on the wire surface. Table 1. The result of the durability test

5

Test subject

Trial # Crack position [mm] # of cycles until crack

w/ soft tube

1 2

10–15 10–15

2100 2450

w/o soft tube 1 2

20–25 15–20

190 170

Deformation Test

It was tested whether the prototype was deformed according to the model based on the PCC assumption. One wire length l2 = L = 50 mm was constant, l1 was determined so that the cap angle θ changed from 0◦ to 90◦ in 15◦ steps, and the wire l1 was pulled by the motor. The deformation was photographed with a calibrated camera, and the cap angle θ and the cap position p was measured from the markers on the base and the cap. Figure 5 shows the deformation. Figure 6 shows the response of the cap angle θ to the wire length displacement Δl1 , the cap position p(x, y) in the task space, and their errors. The cap angle θ increased approximately linearly with the wire length displacement Δl1 , as in the model. However, the magnitude of the error increased in proportion to Δl1 . The relative error was between 10% and 20%. The cap position p(x, y) also drew an arc-shaped trajectory like the model. The error increased in proportion to Δl1 , and was up to about 6 mm. The prototype deformed in almost the same way as the model. Still, the error is about 15%, probably because the PCC assumption is not strictly established. In [8], modeling is performed by balancing the pulling force and the elastic force of the cantilever beam, while assuming PCC. In contrast, in the mechanism proposed above, the wire to be pulled is not an elastic beam, but the base material of silicone resin exerts elastic force, and it is difficult to apply the method of [8]. In [9] and [10], elastic beams are bent by wires, and deformation models considering friction are established without assuming PCC. Although, in the mechanism proposed above, the contracting, sufficiently thick elastic mass is used instead of an elongated elastic beam that does not contract, and its

Wire-Pulling Mechanism with Embedded Soft Tubes for Robot Tongue

319

application is still difficult. More accurate kinematics and dynamics modeling are for further study. Backlash in the Bowden cable mechanism can be considered as another error factor. In this paper, the deformation was measured with the calibrated camera. However, when applied to the tongue mechanism of a vocal robot, it is difficult to measure deformation using a camera. An extension/contraction/bending sensor that can be incorporated into a soft body made of silicone resin is also an issue for the future.

(a) Δl1 = 0 mm, θd = 0 deg

(b) Δl1 = 10.5 mm, θd = 30 deg

(c) Δl1 = 20.9 mm, θd = 60 deg

(d) Δl1 = 31.4 mm, θd = 90 deg

Fig. 5. Photographs of the deformation test. θd means the theoretical value of θ. The yellow line is the neutral line l on the surface of the deformable part. Note that the red line on the surface of the prototype does not match the neutral line in the plane where the wires exist because the prototype does not deform uniformly in Z direction.

Fig. 6. Result of the cap angle θ to the wire length displacement Δl1 , the cap position p(x, y) in the taskspace.

6

Conclusion

Aiming at development of a tongue mechanism that can be mounted on a vocal robot, this paper proposed the wire-pulling mechanism with embedded soft tubes as a method to incorporate wires into soft material. The 2-DOFs input planar prototype was built and the kinematic model under the PCC assumption was formulated. Through the experiment, it was confirmed that the durability was improved as compared with one without soft tubes. In the deformation test, it was confirmed that there was an error of about 15%, although it was

320

N. Endo

deformed approximately according to the model. More accurate modeling and extension/contraction/bending sensors that can be incorporated into soft body are the future studies. Two degrees-of-freedom input is insufficient as a tongue mechanism for a vocal robot. The construction of a deformation model and control system with more degrees-of-freedom input configurations is also an issue for the future.

References 1. Fukui, K., Nishikawa, K., Ikeo, S., Shintaku, E., Takada, K., Takanobu, H., Honda, M., Takanishi, A.: Development of a talking robot with vocal cords and lips having human-like biological structures. In: Proceedings of the 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2005, Canada, pp. 2526–2531 (2005) 2. Yoshikawa, Y., Asada, M., Hosoda, K., Koga, J.: A constructivist approach to infants’ vowel acquisition through mother-infant interaction. Connect. Sci. 15(4), 245–258 (2003). https://doi.org/10.1080/09540090310001655075 3. Takanobu, H., Nakamura, K., Takanishi, A., Ohtsuki, K., Ozawa, D., Ohnishi, M., Okino, A.: Human skull robot as a mechanical patient simulator for mouth opening and closing training. In: Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2001, USA, pp. 2154–2159 (2001) 4. Hofe, R., Moore, R.K.: Towards an investigation of speech energetics using ‘AnTon’: an animatronic model of a human tongue and vocal tract. Connect. Sci. 20(4), 319–336 (2008). https://doi.org/10.1080/09540090802413251 5. Endo, N., Kojima, T., Endo, K., Iida, F., Hashimoto, K., Takanishi, A.: Development of anthropomorphic soft robotic hand WSH-1RII. In: Padois, V., Bidaud, P., Khatib, O. (eds.) Romansy 2019 - Robot Design, Dynamics and Control, vol. 544, pp. 175–182. Springer (2013). https://doi.org/10.1007/978-3-7091-1379-0 22 6. Endo, N., Kojima, T., Ishihara, H., Horii, T., Asada, M.: Design and preliminary evaluation of the vocal cords and articulator of an infant-like vocal robot “Lingua”. In: Proceedings of the 2014 IEEE-RAS International Conference on Humanoid Robots, Humanoids 2014, Spain, pp. 1063–1068 (2014) 7. Webster, R.J., Jones, B.A.: Design and kinematic modeling of constant curvature continuum robots: a review. Int. J. Robot. Res. 29(13), 1661–1683 (2010). https:// doi.org/10.1177/0278364910368147 8. Mishra, A.K., Mondini, A., Del Dottore, E., Sadeghi, A., Tramacere, F., Mazzolai, B.: Modular continuum manipulator: analysis and characterization of its basic module. Biomimetics 3(1), 3 (2018). https://doi.org/10.3390/biomimetics3010003 9. Jung, J., Penning, R.S., Ferrier, N.J., Zinn, M.R.: A modeling approach for continuum robotic manipulators: effects of nonlinear internal device friction. In: Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2011, USA, pp. 5139–5146 (2011) 10. Hsiao, K., Mochiyama, H., Takei, T., Shinotsuka, E., Ogawa, K.: Continuum manipulator with rubber skin layer including pulling-wire mechanism. J. Robot. Soc. Jpn. 35(3), 221–229 (2017). https://doi.org/10.7210/jrsj.35.221

Kineto-static Analysis of a Compact Wrist Rehabilitation Robot Including the Effect of Human Soft Tissue to Compensate for Joint Misalignment Ying-Chi Liu(B) and Yukio Takeda Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan {liu.y.bm,takeda.y.aa}@m.titech.ac.jp

Abstract. Developing a simple, comfortable rehabilitation robot that can carry out in-home rehabilitation has been a long-time challenge. In this paper, we present a rehabilitation robot with one degree of freedom (DOF) for wrist joint flexionextension movement. Passive joints have been added to the exoskeleton, forming a four-bar slider crank mechanism, which can reduce unwanted forces due to joint misalignment. A concept of modeling human soft tissue as a passive prismatic joint with spring is introduced in order to achieve the compactness and comfort of the robot simultaneously. In addition, the effects of human soft tissue displacement are compared. A trade-off between robot volume and comfort is discussed. Finally, the kineto-static analysis of the proposed design is conducted to prove the feasibility of adopting this concept in robot-assisted rehabilitation. Keywords: Robotics · Exoskeleton · Wrist rehabilitation · Joint misalignment · Kineto-static analysis · Human soft tissue

1 Introduction Most daily activities of human beings depend on the exercise capacity of the upper limbs. Patients with impaired upper limb function need rehabilitation to restore their ability to perform activities of daily living (ADL). In rehabilitation, conventional physical therapy is considered as an effective and efficient therapy. However, this approach requires a lot of labor and time, which results in a reduction in the therapeutic effect due to the lack of therapists. On the other hand, in the past four decades, robot-assisted rehabilitation technology has been continuously developed [1]. Norouzi-Gheidari et al. [2] conducted a systematic review and concluded that robot-assisted therapy is an effective method for the restoration of joint motor function. Indeed, the robot can achieve intensive and repetitive treatment for a longer period of time for rehabilitation and assistance, thereby reducing the heavy workload of the therapist. At the same time, through the development of entertainment-assisted technologies related to robots, patients can be better encouraged to participate in rehabilitation exercises in order to obtain better rehabilitation results. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 321–329, 2021. https://doi.org/10.1007/978-3-030-58380-4_39

322

Y.-C. Liu and Y. Takeda

The wrist joint is one of the complex joints of humans. It promotes movement and transmits force through five metacarpal bones, eight carpal bones, radius and ulna. In fact, the human wrist is also a combination of bone and soft tissue including tendons, ligaments, muscles, skin, etc. It connects to the hand from the end of the forearm and performs flexion-extension and radial-ulnar deviation through the muscles of the forearm. In addition, the wrist joint plays a crucial role in independence and is used to perform various tasks in daily life [3]. Therefore, it is important to recover the range of motion (ROM) of the wrist joint through rehabilitation to operate the hand functions. Over the years, there have been a wide variety of robotic devices used for wrist rehabilitation [4–6]. However, in many rehabilitation devices used in the clinical applications, the satisfactory solution cannot be obtained due to some negative feedback [7]. Especially for the exoskeletons, the potential for the misalignment between the robot and the user’s joint would generate unwanted forces that make the user feel uncomfortable or pain results in an unsafe situation during rehabilitation. To solve this problem, our previous works conducted in [8, 9] have analyzed the unwanted force caused by joint misalignment of the wrist rehabilitation robot. With the use of compliant elements and supplementary passive joints, compensation of joint misalignment can be done. Furthermore, to make a portable rehabilitation robot is not only a requirement but also a challenge in recent years. Generally, portability represents the realization of in-home rehabilitation, which allows patients living in rural areas with scarce medical resources to rehabilitate without going to the hospital [10]. In other words, a compact and lightweight robot design is required. In this research, the kineto-static analysis of the wrist rehabilitation robot with the addition of passive joints for joint misalignment compensation is discussed. We introduce the concept in which the human soft tissue is modeled as a passive joint with spring characteristics into the exoskeleton so that the user can wear the exoskeleton without worrying about the unwanted forces due to joint misalignment. Moreover, the robot design may be considered as a comfortable, portable and safer rehabilitation robot that possesses functionality for in-home rehabilitation.

2 Method 2.1 Mechanical Design A conceptual schematic diagram of a wrist rehabilitation robot is illustrated in Fig. 1a. The robot has one degree of freedom (DOF), which can realize the flexion and extension of the wrist joint. The coordinate system is affixed to the human wrist joint, and its origin O is located at the center of the wrist joint. The y-axis points to the hand along the forearm, and the rotation axis of wrist joint is aligned with the x-axis. The links connected to the braces are fastened to the forearm. The human hand is firmly attached to the handle. For flexion-extension movement, the range of motion (ROM) is assigned from −40° to 40°, and the wrist joint angle is 0° when the hand is in a neutral posture.

Kineto-static Analysis of a Compact Wrist Rehabilitation Robot (a)

(b)

R

rigid links braces

z

y

handle

O x

forearm

323

hand

braces Ra P

r3

handle

θ3

r2 θ2 CRhum

r1

wrist joint

z x

y

Fig. 1. (a) Sketch of a wrist rehabilitation robot. (b) Analytical model of a robot.

Figure 1b shows the planar kinematic model composed of the forearm, hand braces and rehabilitation robot, which will be considered in this paper. The point CRhum marked in Fig. 1b is the rotation center of wrist joint. The length between the center of attachment and the center of wrist joint is r 1 . Here, the r 1 value changes during motion, which is related to the displacement of the human soft tissue. The distance from the wrist joint center of rotation to the center of handle is r 2 , which is given as 80 mm. And r 3 represents the length of rigid links connecting the braces and the handle. The following assumptions are made in the analysis: 1. The handle is firmly grasped by the hand, and the hand deformation is neglected. 2. The displacement of the forearm soft tissue is only a translation without rotation and slippage. 3. Velocity related effects are not considered so that we can model the soft tissue of forearm as a spring. Based on the above assumptions, we can model the forearm soft tissue as a passive prismatic joint. Three passive joints are shown in the closed-loop to form a planar RRPR four-bar slider-crank linkage, as shown in Fig. 1b which is able to adapt for joint misalignment. This approach follows our previous study [9] by adding passive joints into the kinematic chain. In Fig. 1b, P, R, and Ra denote prismatic, revolute, and active revolute joints, respectively. It is important to know that no matter how tightly the brace cramped and attached on the forearm is, the soft tissue of the forearm can still move due to its soft nature. When designing a wearable device, this effect cannot be ignored. Although the attributes of human soft tissue are related to the individual’s age, gender, and weight, in this study, we consider the range of displacement in which human soft tissue can move smoothly without pain. We observe the measurement location as a natural length as 80 mm from the center of wrist joint. As a preliminary observation by the authors, we set the amount of tissue expansion and shrinkage movements as 9–10 mm and 8–9 mm, respectively. As a result, the total displacement was 17–19 mm under the condition without any painful sensation. It should be pointed out that in this paper, we only consider the displacement of human forearm soft tissue along the direction of the forearm aligned with the y axis direction. For simplicity, we ignore the displacement and rotation of the human soft tissue in other directions.

324

Y.-C. Liu and Y. Takeda (a)

Ra

forearm

initial/natural posture r3 r2

(b)

P

R

r3 R

hand

θ2

∆s

∆s

CRhum

r1

r1 − ∆s r1

r3

θ2 = 0 °

r2 θ = 40 ° 2

Fig. 2. (a) Model of four-bar linkage. (b) Geometric analysis of four-bar linkage.

Next, we determine the initial distance r 1 between the braces and the wrist joint, which is also regarded as a natural length, as shown in Fig. 2a, and the length of rigid link r 3 . Here, by changing the wrist rotation angle θ 2 between 0° and 40°, only the wrist extension motion is considered. We assumed that the human soft tissue displacements at both θ 2 = 0° and 40° are the same value s. Then, r 3 can be determined at θ 2 = 0° with a given s value as shown in Fig. 2b. Also, by using the following Eq. (1), r 1 can be obtained. For instance, if s and r 3 are given as 5 mm and 80 mm, respectively, r 1 and r 3 were obtained as 81.04 mm and 156.04 mm, respectively. (r1 + s + r2 cos θ2 )2 + (r2 sin θ2 )2 = (r1 − s + r2 )2

(1)

2.2 Kineto-static Analysis For the position analysis of the four-bar linkage from the geometry shown in Fig. 3, by introducing complex number, a vector-loop equation can be written as

Y B

r3 A

θ3 r1

r2

θ2

C

X

Fig. 3. Vector-loop of four-bar linkage.

r3 ejθ3 = r1 + r2 ejθ2 .

(2)

Then, Eq. (2) can be separated into its real and imaginary components, and r 1 and θ 3 can be solved with the quadratic formula and the trigonometric substitution,    −2r2 cosθ2 ± (2r2 cosθ2 )2 − 4 r22 − r32 (3) r1 = 2

Kineto-static Analysis of a Compact Wrist Rehabilitation Robot

θ3 = tan−1



 r2 sin θ2 π π , − ≤ θ3 ≤ r1 + r2 cos θ2 2 2

(4)

−F2y d

A

Tin

Ra ∆x P r1 Fs = − k ∙ ∆x ∆s : ∆xmax

B

R

A

R

−F3y

B −F2x

F3x c

Tout

C

325

Tin F3y

−F3x A

Fs

F4

T4

F2y Tout F1x

B a F2x

C b

F1y

Fig. 4. Free body diagram of four-bar linkage.

Next, in order to analyze the joint force and the input torque required to achieve the desired movement under the wrist joint torque, a static analysis was performed based on the above position analysis. The free body diagram of the linkage for this analysis is shown in Fig. 4. The coupler is input link and we do not consider any inertia force being applied due to the low speed of the rehabilitation motions. In addition, considering the characteristics of soft tissue, we applied the concept of a dynamic pair in [11] and assumed it as a spring attached to the wrist joint. Therefore, the soft tissue of the forearm is considered to have a spring-like behavior. It is important to note that when determining the behavior of the human soft tissue, it is usually regarded as a whole segment without separating the various tissue layers. Obviously, the spring constant of the user’s forearm may change as a non-linear spring element, for instance, it may be increased rapidly with extreme stretching. In this research, we only focus on the smooth movement of soft tissue within a specific deformation range and assume that it satisfies the linear relationship between the applied force and the amount of deformation. The spring constant of the forearm found in the literature could be varied due to different measurement environment and here it is set as 143 N/m [12]. Furthermore, according to biomechanical analysis of daily life activities, we found that the maximum torque of flexion/extension motion is 0.35 N·m [4, 5]. However, for patients in need of rehabilitation, their joints usually have a higher resistance to joint movement due to surgery, inflammation and so on, thus, the output torque used in this study is 1.2 N·m. The parameters used in analysis are shown in Table 1.

326

Y.-C. Liu and Y. Takeda Table 1. The values of parameters used in analysis. Description

Parameter Value

Unit

Distance between brace and wrist joint r 1 Length of hand r2

81.04

mm

80.0

mm

Length coupler link

r3

156.04 mm

Spring constant

k

143

N/m

Output torque

T out

1.2

N·m

3 Results and Discussion In this section, the results of the kineto-static analysis of the wrist rehabilitation device with one degree of freedom (DOF) are discussed. Thanks to the deformation of the soft tissue, the forearm tissue can be regarded as a prismatic joint moving along the limb. This concept facilitates the compact design of the robot, allowing a lightweight structure, which is desirable for portability. We should know that the stroke of the prismatic joint must be carefully determined. Indeed, the choice of forearm soft tissue displacement has a great influence on robot design. Human soft tissue is a complex material, and its maximum expansion and contraction displacement may be related to individual differences. In this research, the total soft tissue movement ranged from 6 to 14 mm within the preliminary observation range of maximum forearm soft tissue displacement. For wrist extension, the wrist angle ranges from 0° to 40°, and the length of the hand is fixed as 80 mm. The initial position of the braces r 1 and the length of the rigid link r 3 can be obtained by Eq. (1). The calculation results of r 1 and r 3 for each selected displacement s are shown in Fig. 5. The results show that a larger forearm displacement leads to a longer link length and the farther position of the braces. First, the longer rigid link length increases the weight and volume of the exoskeleton, which causes adverse effects on portability. Taking these factors into account, it is required that the length of the rigid link does not to exceed 200 mm, which means that the case of 2s = 12 mm and 14 mm soft tissue displacement may not be appropriate. Also, the initial position of the forearm’s braces approaches to the wrist joint when the forearm displacement decreases. However, it is generally considered that the tissue near the wrist is hard, and it should be avoided that the attachment is close to the wrist to affect the comfort of the wearer. We expect the location of the braces to be more than 50 mm from the wrist. Then, we can say that the case of 6 mm and 8 mm soft tissue displacement may not be suitable. As a result, the soft tissue displacement of 10 mm is selected, which is the most suitable design for a comfortable and portable design. It is true that 10 mm is about half of the allowable range of human soft tissue, which is around 20 mm. It can ensure that the motion of human soft tissue can be smooth. Kinematic analysis of the robot designed above has been done by changing the crank angle θ 2 between 0° and 40°. Using the equations in Sect. 2, the coupler angle θ 3 and the slider position r 1 are obtained as shown in Fig. 6. The dotted line indicates the initial position of the brace at 81 mm from the wrist joint. During the wrist extension motion,

Kineto-static Analysis of a Compact Wrist Rehabilitation Robot

327

Fig. 5. Effect of varying human soft tissue displacement.

the distance from the center of the wrist joint to the center of the brace changes from 76 mm to 86 mm due to the movement of the soft tissue of the forearm.

Fig. 6. Result of kinematic analysis.

(a)

(b)

Fig. 7. Result of kineto-static analysis.

328

Y.-C. Liu and Y. Takeda

The static analysis results are shown in Fig. 7. As we can observe from Fig. 7a, the resultant force applied on the wrist joint is within the maximum force 20 N. The force in the y-axis direction is much larger than that in x. In other words, during wrist extension motion, the exerted force is mainly applied in the direction of perpendicular to the forearm. The magnitude of this force increases as the rotation angle of the wrist increases. Also, in Fig. 7b, the input torque was obtained to generate the constant desired output torque. Consequently, the input torque varied from 2.34 to 2.94 N·m which is for determining the motor design parameters.

4 Conclusion This paper has discussed the kineto-static analysis of the wrist rehabilitation robot, which provides effective ideas for designing a more comfortable and portable rehabilitation robot. First, the addition of passive joints into the exoskeleton was applied to compensate for the unwanted force due to joint misalignment. A concept of modeling human soft tissue as a passive prismatic joint was applied. Thanks to the deformation of human soft tissue, one prismatic joint in a slider-crank linkage can replace the soft tissue effect. Compared with other existing devices, this approach provides an effective design in compactness and portability. Furthermore, a comparison of the magnitude of human soft tissue displacement was discussed in order to minimize the size of the exoskeleton without affecting the comfort of users. Finally, the kineto-static analysis was conducted to show the feasibility of the current design. Due to its portability and ability to compensate for joint misalignment, we expect that the proposed design can be used for portable and safer in-home rehabilitation. Acknowledgments. This research was in part supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science (17H03162), and by ABE TECHNO SYSTEM Co., Ltd.

References 1. Maciejasz, P., Eschweiler, J., Gerlach-Hahn, K., Jansen-Troy, A., Leonhardt, S.: A survey on robotic devices for upper limb rehabilitation. J. Neuroeng. Rehabil. 11(1), 3–31 (2014) 2. Norouzi-Gheidari, N., Archambault, P.S., Fung, J.: Effects of robot-assisted therapy on stroke rehabilitation in upper limbs: Systematic review and meta-analysis of the literature. J. Rehabil. Res. Dev. 49(4), 479–496 (2012) 3. Ryu, J., Cooney, W.P., Askew, L.J., An, K.-N., Chao, E.Y.S.: Functional ranges of motion of the wrist joint. J. Hand Surg. Am. 16(3), 409–419 (1991) 4. Pezent, E., Rose, C.G., Deshpande, A.D., O’Malley, M.K.: Design and characterization of the openwrist: A robotic wrist exoskeleton for coordinated hand-wrist rehabilitation. In: 2017 International Conference on Rehabilitation Robotics (ICORR), pp. 720–725 (2017) 5. McDaid, A.J.: Development of an Anatomical Wrist Therapy Exoskeleton (AW-TEx). In: 2015 IEEE International Conference on Rehabilitation Robotics (ICORR), pp. 434–439 (2015)

Kineto-static Analysis of a Compact Wrist Rehabilitation Robot

329

6. Singh, N., Saini, M., Anand, S., Kumar, N., Srivastava, M.V.P., Mehndiratta, A.: Robotic exoskeleton for wrist and fingers joint in post-stroke neuro-rehabilitation for low-resource settings. IEEE Trans. Neural Syst. Rehabil. Eng. 27(12), 2369–2377 (2019) 7. Näf, M.B., Junius, K., Rossini, M., Rodriguez-Guerrero, C., Vanderborght, B., Lefeber, D.: Misalignment compensation for full human-exoskeleton kinematic compatibility: State of the art and evaluation. Appl. Mech. Rev. 70(5), 1–19 (2019) 8. Liu, Y.-C., Takeda, Y.: Static analysis of a wrist rehabilitation robot with consideration to the compliance and joint misalignment between the robot and human hand. In: Proceedings of Annual Conference of the Robotics Society of Japan 2019, Tokyo (2019) 9. Liu, Y.-C., Takeda, Y.: Kineto-static analysis of a wrist rehabilitation robot with compliant elements and supplementary passive joints to compensate the joint misalignment. In: The 25th Jc-IFToMM Symposium, Japan (2019) 10. Xiao, Z.G., Menon, C.: Towards the development of a portable wrist exoskeleton. In: 2011 IEEE International Conference on Robotics and Biomimetics, pp. 1884–1889 (2011) 11. Takeda, Y., Sugahara, Y., Matsuura, D., Matsuda, S., Suzuki, T., Kitagawa, M., Liu, Y.-C.: Introduction of dynamic pair to modeling and kinemato-dynamic analysis of wearable assistdevices. In: The JSME Annual Mechnical Engineering Congress 2019, Akita, Japan (2019) 12. Yu, T.F., Wilson, A.J.: A passive movement method for parameter estimation of a musculoskeletal arm model incorporating a modified hill muscle model. Comput. Methods Programs Biomed. 114(3), e46–e59 (2014)

A Mobile Robot Which Locomotes on Walls to Interact with Rodents Soichi Yamada1 , Keitaro Ishibashi2 , Hiroya Yokoyama2 , Jiei Yanagi2 , Atsuo Takanishi3 , and Hiroyuki Ishii3(B) 1 School of Creative Science and Engineering, Waseda University, Tokyo, Japan 2 Graduate School of Creative Science and Engineering, Waseda University, Tokyo, Japan 3 Faculty of Science and Engineering/Humanoid Research Institute, Waseda University, Tokyo

1698555, Japan [email protected]

Abstract. Rodents are often used in medicine, brain science, pharmacology, and psychology, as experimental animals. We have been proposing a novel method that involves usages of mobile robots as agents to interact with animals to understand animal behavior. A robot which is able to locomote not only on floors but also walls are strongly expected to increase the variety of behavioral experiments. The purpose of this study is to develop a new mobile robot which is able to locomote both on floor and on walls interacting with rats. For this purpose, we invented a novel mechanism which made a mobile robot locomote by wheels on iron walls. The mechanism consists of an active bending joint in the body, a passive tilting magnet, and a passive tilting tail. We implemented this mechanism into a new small mobile robot, named WR-7 (Waseda Rat No. 7). In this paper, proposed mechanism and its implementation are described. Keywords: Mobile robot · Locomotion on wall · Magnetic attraction · Interaction with animals · Mechanical design

1 Introduction Rodents, such as rats and mice, are often used in medicine, brain science, pharmacology, and psychology, as experimental animals, and behavioral experiments have been done in these fields [1–5]. We have been developing several mobile robots, which interact with rats, and have been proposing a novel method that involves usages of these robots as agents to interact with animals to understand animal behavior [6–8]. Usages of robots in behavioral experiments on animals becomes popular and contribute for better understanding of animal behavior [9–11]. However, locomotion abilities of our conventional robots are limited comparing with real rats. For instance, none of them are able to locomote on walls while rats often climb walls to play with other rats. Therefore, the robot which is able to locomote not only on floors but also walls are strongly expected to increase the variety of behavioral experiments. The purpose of this study is to develop a new mobile robot which is able to locomote both on floor and on walls interacting © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 330–337, 2021. https://doi.org/10.1007/978-3-030-58380-4_40

A Mobile Robot Which Locomotes on Walls to Interact with Rodents

331

with rats. We invented a novel mechanism which made a mobile robot locomote with wheels on iron walls. We then implemented this mechanism into a new small mobile robot, named WR-7 (Waseda Rat No. 7). In this paper, proposed mechanism and its implementation into WR-7 are described.

2 Mechanism for Locomotion Both on Floor and Wall 2.1 Target Environment; Box for Interaction Experiment Interaction experiments between the mobile robot and rats are conducted in a box consisting of iron walls and a plastic floor as shown in Fig. 1. Interactions between the robot and the rats occur both on the floor and on the walls in this box. The walls should be climbable not only for the robot but also for rats. Rats can climb walls with rough surfaces using their crows, while they can not climb those of flat surface with few concavities and convexities. Therefore, the walls should not be just flat iron boards and should involves concavities or convexities. An iron board with many small square holes, which were 15 × 15 mm, was then employed in this study. This board is made of a coldreduced carbon steel sheet of 1 mm thickness. The floor should be flat for locomotion of the robot and the rats. For transfers from the floor to the walls, the floor and the walls are perpendicularly placed. In addition to that, floor should be made of non-magnetic material. Therefore, the floor of the box is made of plastic.

Fig. 1. Wall made of iron. A rat is walking on the wall.

2.2 Locomotion on Wall by Magnetic Attraction There are several methods to make mobile robots locomote on walls, such as climbing by legs, hanged by tethers, or moving by wheels with supports by magnetic attractions. In this study, moving by wheels with supports by magnetic attractions were employed as a method to make a mobile robot locomote on walls. A robot with magnets can hold its position on a wall made of iron. Magnets on the robot generate attaching force on the walls, and these attaching forces generate friction forces between the robot and the

332

S. Yamada et al.

walls. Based on the principle of the coulomb friction, these friction forces are increased by increasing attaching forces, which are generated by magnetic attractions. Therefore, intensity of magnetic attractions should be designed to make the robot locomote on the wall. If magnetic attraction is too small, the robot is not able to hold its body on the wall during locomotion. If magnetic attraction is too large, the robot is not able to locomote on the wall because of large friction forces. 2.3 Arrangement of Magnets Arrangement of magnets on the robot should be decided considering relationships between attaching forces generated by magnetic attractions and friction forces generated by these attaching forces. Embedding magnets on the body of the robot is the simplest design approach to obtain attaching forces on the walls made of iron. However, this approach involves limitations in the stability of attaching forces. Embedding magnets in wheels is another common design approach to increase both mobility and stability on the walls made of iron. Therefore, a design approach to implement magnets both in the body and in wheels was employed in this study. All the forces applied on the robot, during locomotion on the walls, are shown in Fig. 2. In addition, the robot should have functions to transfer smoothly between the floor and the walls. As described above, the floor and the walls are perpendicularly placed. Two different mechanisms were then invented for smooth transfer between the floor and the walls. A passive tilting magnet mechanism for the transfer from the floor to the walls, as shown in Fig. 3, was invented. This mechanism consists of a magnet in a case and a passive rotational joint which enables the case to make passive rotations on the rear body. A passive tilting tail mechanism for the transfer from the walls to the floor, as shown in Fig. 4, was also invented. This mechanism consists of a tail made by rigid material and a passive rotational joint.

Fig. 2. Mechanism to locomote on walls with magnetic attraction (left) and its free body diagram (center and right)

A Mobile Robot Which Locomotes on Walls to Interact with Rodents

333

3 Mechanical Design of Mobile Robot 3.1 Feature and Specifications A photo of WR-7 is shown in Fig. 5. The size of the robot is almost equal to a mature rat. All the components of the robot, including an onboard control system, wireless

Fig. 3. Mechanism of passive tilting magnets (left). The active bending joint rift the front body when the robot is approaching to the wall, and the passive tilting magnet is then pulled from the wall by magnetic attraction (center). Once the robot starts climbing the wall, the active bending joint moves the rear body to touch the wall (right).

Fig. 4. Mechanism of passive tilting tail (left). The passive tilting tail touches the floor to obtain reaction force F GND when the robot moves down (center). The rear body moves away from the wall by the moment generated by F GND and Mg when the active bending joint starts bending the body. The active bending joint starts stretching the body when the wheel touches the floor to move away from the wall (right).

334

S. Yamada et al.

communication module and battery, are implemented on the robot. Thus, this robot can locomote by remote control from a PC. Locomotion velocity of the robot on the floor and the wall are 0.24 m/s. 3.2 Mechanical Design The mechanical deign is shown in Fig. 6. Two active wheels of the robot are respectively driven by two geared DC motors (RE16, 4.5 W, Maxon Motor) with rotary encoders.

H:85

Unit: mm

z x

y

Fig. 5. Feature of WR-7

Supporting point

Battery Servo motor for bending body Onboard control system

Wheel wit magnet

Passive tilting magnet

Passive tilting tail

DC motors Fig. 6. Mechanical Design of WR-7

Supporting point

A Mobile Robot Which Locomotes on Walls to Interact with Rodents

335

Each motor is connected with right/left wheel via bevel gears. The active bending joint is driven by a servo motor (KRS-3304, Kondo). A neodymium magnet (50 × 10 × 5 mm, 19 g) is embedded in a tilting case which is hanged from the body. 10 neodymium magnets (10 × 10 × 3 mm) are embedded in the wheels like a snow chain for automobiles. The wheels, body, tilting tail and tilting case for magnets are made by 3D printer (ProJet 3600 HDMAX, 3D systems). Several parts for driving, such as shafts and gears are made of irons, and gearboxes is made of aluminum alloy. 3.3 Onboard Control System and Its Electric Configuration A microcontroller (STM32Nucleo-F303K8, STmicroelectronics), wireless communication module (XBee Pro), Li-Po battery, and motor drives are implemented on the robot. Position of each DC motor is controlled by the microcontroller via the motor driver using a PWM module and a hardware counter in the microcontroller. Position of the servo motor is controlled by embedded controller in it according to the instructions from the microcontroller. Motions of these motors are controlled by a PC via wireless communication. Operation time of the robot with full charged battery is 40–60 min.

4 Performance of Mobile Robot 4.1 Locomotion on Wall An experiment to confirm locomotion performance of WR-7 on the wall was performed in the box as described in Chapter 2. In this experiment, the robot was controlled via wired communication for stable operations. As shown in Fig. 7, the robot locomoted on the wall without falling down. It moved forward and back on the wall, and turned around on the wall too.

t=0s

t=1s

t=2s

Fig. 7. WR-7 moved on the wall made by iron. The robot started turning on the wall at t = 0 s and stopped at t = 1 s. The robot then started moving back on the wall at t = 2 s.

336

S. Yamada et al.

4.2 Transfer Between Floor and Wall An experiment to confirm transfer performance of WR-7 between the floor and the wall was performed in the box. As shown in Fig. 8, the robot moved from the floor to the wall smoothly. As shown in Fig. 9, the robot moved from the wall to the floor smoothly too.

t=0s

t=2s

t=4s

Fig. 8. WR-7 moved on the floor to approach to the wall. The robot reached to the wall at t = 0 s. The robot moved to the wall from the floor at t = 2 s. The robot then started moving on the walls at t = 4 s.

t=0s

t=3s

t=6s

Fig. 9. WR-7 moved on the wall to approach to the floor. The robot reached to the floor at t = 0 s. The robot moved to the floor from the wall at t = 3 s. The robot then started moving on the floor at t = 6 s.

A Mobile Robot Which Locomotes on Walls to Interact with Rodents

337

5 Conclusion We invented a novel mechanism which made a mobile robot locomote by wheels on iron walls. A new mobile robot, named WR-7 (Waseda Rat No. 7), which was able to locomote both on floor and on walls interacting with rats, was then developed. Locomotion performances of WR-7 were confirmed in the experiments. WR-7 is expected to be used in interaction experiments between it and rats.

References 1. Van der Staay, F.J.: Animal models of behavioral dysfunctions: basic concepts and classifications, and an evaluation strategy. Brain Res. Rev. 52, 131–159 (2206) 2. Louis, C., Stemmelin, J., Boulay, D., Bergis, O., Cohen, C., Grieble, G.: Additional evidence for anxiolytic- and antidepressant- like activities of saredutant (SR48968), an antagonist at neurokinin-2 recepter in various rodent-models. Pharmacol. Biochem. Behav. 89, 36–45 (2008) 3. Kelly, J.P., Wrynn, A.S., Leonard, B.E.: The olfactory bulbectomized, rat as a model of depression: an update. Pharmacol. Ther. 74, 299–316 (1997) 4. Zalocusky, K.A., Ramakrishnan, C., Lerner, T.N., Davidson, T.J., Knutson, B., Deisseroth, K.: Nucleus accumbens D2R cells signal prior outcomes and control risky decision-making. Nature 531, 642–646 (2016) 5. Michael, X., Barrus, M., Winstanley, C.A.: Dopamine D3 receptors modulate the ability of win-paired cues to increase risky choice in a rat gambling task. J. Neurosci. 36(3), 785–794 (2016) 6. Ishii, H., et al.: Experimental study on task teaching to real rats through interaction with a robotic rat. In: Nolfi, S., et al. (eds.) SAB 2006. LNCS (LNAI), vol. 4095, pp. 643–654. Springer, Heidelberg (2006). https://doi.org/10.1007/11840541_53 7. Ishii, H., et al.: A novel method to develop an animal model of depression using a small mobile robot. Adv. Robot. 27(1), 61–69 (2013) 8. Li, C., et al.: Bioinspired phase-shift turning action for a biomimetic robot. IEEE/ASME Trans. Mechatron. 25(1), 84–94 (2020) 9. Halloy, J., et al.: Social integration of robots into groups of cockroaches to control selforganized choices. Science 18, 1155–1158 (2007) 10. Gribovskiy, A., Halloy, J., Deneubourg, J.-L., Bleuler, H., Mondada, F.: Towards mixed societies of chickens and robots. In: Proceedings of the 2010 IEEE international Conference on Intelligent Systems and Automations (2010) 11. Bonnet F., et al.: Robots mediating interactions between animals for interspecies collective behaviors. Sci. Robot. 4 (2019)

Development of Small Robot with Inline Archimedean Screw Mechanism that Can Move Through Wetlands Ko Matsuhiro1(B) , Katsuaki Tanaka1 , Shou Inoue1 , Tingting Zhong1 , Kazuki Kida1 , Yusuke Sugahara3 , Atsuo Takanishi1,2 , and Hiroyuki Ishii1,2 1 Waseda University, Tokyo, Japan

[email protected] 2 Humanoid Robotics Institute, Waseda University, Tokyo, Japan 3 Tokyo Institute of Technology, Tokyo, Japan

Abstract. We developed a small robot with Inline Archimedean Screw Mechanism in response to the demands of surveying Invasive Alien Species (IAS). There are some harmful IAS in wetlands such as snapping turtles. The current method for the survey is net-trap (“Mondori-wana”) and fumbling search by humans. Both methods are inefficient and unsafe for humans. To improve the efficiency and safety of surveying IAS in wetlands, the mobile robot system is demanded. The target environment is configured with soft ground, covered with water, dense water plants and side hole. There is no mobile robot that can move in such an environment. We proposed Inline Archimedean Screw Mechanism to realize locomotion performance at the target environment. The robot can move on a soft land as if cutting a screw. Furthermore, pushing aside water plants, the robot can prevent itself from getting stuck by dense water plants. Using Inline Archimedean Screw Mechanism, we developed the robot, named WANGOT (WAseda Native species Guardian robOT). WANGOT has the size of W85 × H85 × L460 [mm], the weight of 1.4 [kg] and two actuators and four Archimedean screws. It was possible to move forward at a speed of 27 [mm/s] on artificial turf, and the movement performance on the environment of simulated dense water plant area was confirmed. Keywords: Mobility · Archimedean screw · Invasive Alien Species · Wetlands

1 Introduction 1.1 Damage by Invasive Alien Species In recent years, the impact of Invasive Alien Species (IAS) has become serious. Invasive Alien Species can cause three main types of damage; destruction of ecosystems, threat to human safety, economic loss to agriculture, forestry and fisheries. Therefore, controlling the number of IAS is urgent. Snapping turtles are one of the most notable IAS in Japan. Snapping turtles were originally introduced as pets from the American continent but some were discarded into the wild due to difficulty of keeping [1]. In Lake Inba, Chiba © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 338–346, 2021. https://doi.org/10.1007/978-3-030-58380-4_41

Development of Small Robot with Inline Archimedean Screw Mechanism

339

Prefecture in Japan, reproduction of Snapping turtles is getting a serious problem. They are destroying native ecosystem of Lake Inba, and sometimes they attack to human. The main capturing method is the net-like trap called “Mondori-wana”, which is submerged in water and has an irreversible entrance shown in Fig. 1a. However, this trap is less effective during the cold season because the activity level of turtles decreases. For the alternative method, the fumbling survey by human shown in Fig. 1b is performed. In this way, the captor enters the wetlands directly where the water level is around knees to waist, then put arms in the poor visibility water and grope to find turtles. Working on this work has a huge burden for the captor and has the risk of injured by turtles. Even several captors are working on this problem, the number of control turtles has not reached the targeted value to decrease the number of turtles because it takes long time to find them.

Fig. 1. (a) Left: Net-like trap for snapping turtles named “Mondori-wana” (b) Right: Fumbling survey of snapping turtles at Lake Inba

Therefore, using small mobile robot to collect locomotion information of turtles is a one way to solve this problem. 1.2 Objective The objective of this work was to develop the small mobile robot that can collect the locational information of IAS, especially snapping turtles in wetland area. For the first step, we worked on developing small robot with enough mobility in habitat environment of IAS. 1.3 Target Environment The target environment is shown in Fig. 2. The ground is soft soil, covered with water at the level of 0.5 [m] to 1.25 [m]. Some area is filled with dense water plants. Snapping turtles are usually habitat under water plants or side hole at the radius of 100 [mm] to 300 [mm]. Considering target environment and usage for captor, requirements of the robot are as follows.

340

(1) (2) (3) (4)

K. Matsuhiro et al.

Mobility of soft ground Mobility of dense area of water plants Size: W100 × H100 × L500 [mm] Weight: 3 [kg] or less

Fig. 2. Target environment

2 Mechanism 2.1 Locomotion Mechanism Many field robots which have high mobility in the outdoor environment have been developed. WAMOT [2] and Quince [3] are well-known field robots which have high mobility in outdoor environment. Each have high advantage on the complex terrain such as forest or disaster area of earthquake. However, our target environment has dense area of water plants and the robot need to get into the area to reach the side hole, so these robots can’t achieve our requirement because they can’t push aside the water plants. Considering AUV (Autonomous Underwater Vehicle) such as REMUS [4], dense water plants will prevent its propulsion. Therefore, previous robots don’t have enough mobility in our target area of this study. Considering several types of mechanism, we focused on Archimedean screw utilizing the strength of moving on soft ground. Conventionally, Archimedean screw is known as a machine used for transferring water. Otherwise, it has been also applied to vehicles which have high mobility in a complex surface such as snow, mud, sand, or marsh. [5, 6], These are equipped with dual Archimedean screw units, the vehicles can get propulsion from soft land. Kenji et al. [7] shows simplified dynamic model of exploration robot for soft soil equipped with dual Archimedean screw units. In this mechanism, robot can get propulsion by pushing back the soil and prevent to move aside by rotating opposite directions. However, this method has low movement performance in dense area of water plants (Fig. 3).

Development of Small Robot with Inline Archimedean Screw Mechanism

341

Fig. 3. Marsh Screw Amphibian (MSA) [5]

2.2 Inline Archimedean Screw Mechanism In order to realize enough mobility in dense water plants area, we thought that if the shape of the robot is elongated like a snake, it would be able to meet the requirements. Therefore, we put four Archimedean screws inline instead of putting two screws side by side. Inline Archimedean Screw Mechanism consists four Archimedean screws, top and end screws have right spiral (Spiral Screw R-A and Spiral Screw R-B), and two middle screws has left spiral (Spiral Screw L-A and Spiral Screw L-B). Rotational direction and dynamic model of Inline Archimedean Screw Mechanism shown in Fig. 4 and Fig. 5. In Fig. 4, the following equation hold:

Fig. 4. Rotational direction of Inline Archimedean Screw Mechanism (a) Left: x-y plane, (b) right: y-z plane

MR = Fy1 r + Fy4 r

(1)

ML = Fy2 r + Fy3 r

(2)

MR = MR , ML = ML

(3)

Mx = ML − MR ∼ =0

(4)

where MR , ML are moment given to the screws by the motor, MR , ML are reaction moment, Mx is moment around x-axis, Fy1 , Fy2 , Fy3 , Fy4 are y-axis direction of propulsive force of each screws and r is screw radius. Equation (4) refers to the moment around x-axis

342

K. Matsuhiro et al.

is canceled out using two motors located side by side. In Fig. 5, the following equation hold: max = Fx1 + Fx2 + Fx3 + Fx4

(5)

    may = Fy1 + −Fy2 + Fy3 + −Fy4 ∼ =0

(6)

    3 1 1 3 Mz = − Fy1 L + Fy2 L + − Fy3 L + Fy4 L ∼ =0 2 2 2 2

(7)

Where m is weight of a robot, ax , ay are acceleration of x-axis direction and y-axis direction, Fx1 , Fx2 , Fx3 , Fx4 are x-axis direction of propulsive force of each screws and L is distance between screws next to each other. Equation (5) refers to the propulsion force if the y-axis direction is canceled out and Eq. (6) refers to the moment around the y-axis is canceled out using four Archimedean screws. As shown in Eq. (4), (6), and (7), this mechanism cancels out three forces: the moment around the x-axis, the propulsion force in the y-axis direction, and the moment around the z-axis. Achieving three conditions, the robot can move forward. If only two Archimedean screw is used, for example use only spiral screw R-A and spiral screw L-A, Eq. (7) doesn’t hold and robot will make a swivel turn.

Fig. 5. Dynamic model of Inline Archimedean Screw Mechanism: x-y plane

2.3 WANGOT (WAseda Native Species Guardian RobOT) Using Inline Archimedean Screw Mechanism, we developed a small mobile robot named WANGOT shown in Fig. 6. Specifications are shown in Table 1. The screws are driven from the motors through the pinion gears and internal gears. Needle bearings outside the cylindrical housing allow the outer Archimedean screws to rotate. Power is supplied externally. Also, the unit where the motors are located is sealed with an O-ring, and the power supply line is passed through a tube and through a waterproof connector to make the robot waterproof. Detail mechanism of WANGOT is shown in Fig. 7.

Development of Small Robot with Inline Archimedean Screw Mechanism

343

Table 1. Specifications of WANGOT Item Size

Specifications mm3

W85 × H85 × L460

Weight kg

1.4

Number of actuators

2

Motors

12 V SparkFun ROB-12285

Battery

11.1 V 600 mAh

Fig. 6. WANGOT (Waseda Native species Guardian robOT)

Fig. 7. Detail mechanism of WANGOT

3 Experiments and Results Evaluation experiments were conducted to confirm the movement performance of WANGOT. 3.1 Experiment of Movement Ability on Artificial Turf In order to verify the function of Inline Archimedean Screw Mechanism, following experiment conducted. Move the robot 500 [mm] forward on artificial grass (grass height 30 [mm]) and measure the following parameters. The measurement is performed 10 times. • Speed: Calculate the speed from the time required to advance 500 [mm], and compare the average value with the theoretical value. • Straightness (Gap): Measures the deviation to the left and right when moving forward 500 [mm] from the start • Propulsion: Attach a force gauge to the back of the robot to measure the propulsion and calculate the average value. This experiment is performed separately from the above parameters.

344

K. Matsuhiro et al.

Table 2 shows the result. The theoretical value of forward speed is 35 [mm/s] using the pitch of the Archimedean screw is 60 [mm] and 70 [%] output of the gear motor. However, the actual value was 27 [mm/s]. The reason is considered that motor was under heavy load, and slip makes propulsion loss. The Gap occurred 35 [mm]. This is because of motor unit located the back of the robot make additional friction to spiral screw R-B (Fig. 8). Table 2. Movement result of experiment of movement ability on artificial turf Speed mm/sec

27

Gap mm

35

Propulsive force N 15.8

Fig. 8. Experiment of movement ability on artificial turf

3.2 Experiments of Mobility on Artificial Grass with Obstacles In order to confirm the movement performance of a dense area of water plants, an experiment was performed by inserting an ABS rod into artificial turf. ABS rods with a diameter of 3 [mm] and a height of 100 [mm] were used. As shown in Fig. 9, five ABS rods were arranged at intervals of 20 [mm]. The robot moves between the ABS bars. The tensile force of the ABS rod used in the experiment was about 2.6 [N] on average at the point of 50 [mm] from the ground. The experiment was performed 10 times and evaluated by whether the robot was able to break through to the end. As a result, robot could pass through nine times out of ten times. Figure 10 shows robot movement during experiments on mobility on artificial grass with obstacles. Through the experiment, it was confirmed that the ABS rod was pushed out obliquely rearward with respect to the x-axis traveling direction by rotating the Archimedean screw. However, when ABS rod completely enters the gap between screws, robot got stuck.

Development of Small Robot with Inline Archimedean Screw Mechanism

345

Fig. 9. Experiments on mobility on artificial grass with obstacles

Fig. 10. Robot movement during experiments on mobility on artificial grass with obstacles

4 Conclusion In this study, we developed a small robot named WANGOT that can move through wetlands in response to the demands of surveying Invasive Alien Species (IAS). We suggested novel mechanism using Archimedean screw named Inline Archimedean Screw Mechanism. The robot can move on a soft land as if cutting a screw and push aside water plants to prevent itself from getting stuck by dense water plants. Evaluation experiments shows that it was possible to move forward at a speed of 27 [mm/s] on artificial turf, and the movement performance on the environment of simulated dense water plant area was confirmed. For the future works, adding turning mechanism to change direction of the robot should be considered, and we already designed bending mechanism on the robot. Acknowledgement. This study was conducted at the Waseda Research Institute for Science and Engineering, the Humanoid Robotics Institute, and the Future Robotics Organization, Waseda University. This research is supported by the Consolidated Research Institute for Advanced science and Medical Care, Waseda University (ASMeW), SolidWorks K.K., Leading Graduate Program

346

K. Matsuhiro et al.

in Science and Engineering, Waseda University from MEXT, Japan, and Grant-in-Aid for JSPS Research Fellow. Chiba Biodiversity Center supported our field survey and Lake Inba.

References 1. Toshikazu, M., Tetsuro, U.: Invasive alien species in Japan: the status quo and the new regulation for prevention of their adverse effects. In: International Research Initiatives for Environmental Studies, pp. 171–191 (2004) 2. Katsuaki, T., et al.: Design of operating software and electrical system of mobile robot for environmental monitoring. In: International Conference on Robotics and Biomimetics, pp. 1763–1768 (2014) 3. Keiji, N., et al.: Redesign of rescue mobile robot Quince. In: Proceedings of the IEEE Symposium on Safety, Security, and Rescue Robotics (SSRR), pp. 13–18 (2011) 4. Ben, A., et al.: REMUS: a small, low cost AUV; system description, field trials and performance results. In: Oceans 1997. MTS/IEEE Conference Proceedings, vol. 2, pp. 994–1000 (1997) 5. Rush, E.S., et al.: Evaluation of the Marsh Screw Amphibian (MSA). Technical report A-77-1: Evaluation of herbicide application platforms for use in aquatic plant control, Report 1 (1977) 6. Kenji, N., et al.: Terramechanics-based propulsive characteristics of mobile robot driven by Archimedean screw mechanism on soft soil. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4946–4951 (2010) 7. Kenji, N., et al.: Study on soil-screw interaction of exploration robot for surface and subsurface locomotion in soft terrain. Graduate University for Advanced Studies (Sokendai) (2011)

Static Force Analysis of an Omnidirectional Mobile Robot with Wheels Connected by Passive Sliding Joints Tatsuro Terakawa

and Masaharu Komori(B)

Kyoto University, Kyoto Daigaku-katsura, Nishikyo-ku, Kyoto 6158540, Japan [email protected]

Abstract. To solve problems of conventional mobile robots, such as limited mobility and complicated structure, we proposed an omnidirectional mobile robot named slidable-wheeled omnidirectional mobile robot (SWOM). SWOM has three wheels connected by passive sliding joints. The relative movements of the wheels enable SWOM to make an omnidirectional movement. The wheels of SWOM are all normal wheels, so SWOM establishes both superb mobility and a simple structure. In the movement of SWOM, the steering angles and the relative positions of the wheels are varied, which can affect the motion characteristics of SWOM. This paper attempts to evaluate such effects quantitatively based on the statics. This paper describes the structure and movement of SWOM first. Then, a static force analysis is carried out and the characteristics of the input-output relation are discussed considering the situations under the driving force limitation and the friction force limitation. Through the analysis, the conditions that SWOM can gain large output force are clarified. Keywords: Omnidirectional mobile robot · Wheel · Kinematics · Statics

1 Introduction Mobile robots are widely used in a variety of fields, including industry and logistics. Conventional mobile robots that equip normal wheels (e.g., tired wheels) can travel forward/backward and turn. Then, they perform switchbacks or turnabouts to move sideways, but it costs extra time and space. To solve the limited mobility, several wheel mechanisms for omnidirectional mobile robots, which can immediately move in any direction, have been proposed. One example is an omni wheel, which has freely rotating rollers around the wheel main body [1–3]. The mobile robots equipped with more than three omni wheels can make an omnidirectional movement. An active omni wheel, which can actively drive the rollers, has also been proposed [4, 5]. Another example is a spherical wheel, which has a ball-shaped wheel that is rotatable in any direction [6–8]. These specialized wheel mechanisms can realize omnidirectional mobility, but have problems due to the complicated structure. Then, the novel omnidirectional wheel mechanism that has a simple structure is required. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 347–354, 2021. https://doi.org/10.1007/978-3-030-58380-4_42

348

T. Terakawa and M. Komori

To meet this requirement, the authors have proposed a wheel mechanism named slidable wheel and a mobile robot using it [9, 10]. The slidable-wheeled omnidirectional mobile robot (SWOM) has three wheels connected by passive sliding joints. The relative movements of the wheels realize an omnidirectional movement. The wheels are all normal wheels, so SWOM can establish both superb motility and a simple structure. In this paper, we discuss and evaluate the motion characteristics of SWOM based on the statics. The structure and movement of SWOM are explained first. Then, the input-output relation of the static force is theoretically analyzed.

2 Proposed Omnidirectional Mobile Robot We explain the structure of SWOM. As shown in Fig. 1, SWOM consists of one base block, three sliding joints, and three driving units. Three sliding joint rails are fixed to the base block radially at equal intervals. The base block and three rails construct the main body of SWOM. Each driving unit has a wheel, a sliding joint slider, a driving motor, and a steering motor. The driving motor rotates the wheel around its axle, whereas the steering motor rotates the wheel on the ground contact point around the vertical axis. The sliding joint is a passive joint that allows the distance between the base block and the driving unit to vary according to the driving unit travel. Base block Main body Sliding joint rail Sliding joint slider Steering motor Driving motor Driving unit Fig. 1. Proposed omnidirectional mobile robot, SWOM.

Figure 2 illustrates an example of the omnidirectional movement of SWOM. Here, the traveling direction of each wheel is perpendicular to the sliding joint rail, but SWOM can move in the longitudinal and lateral directions and turn thanks to the relative movements

Fig. 2. Omnidirectional movement of SWOM when the moving direction of each wheel is perpendicular to the sliding joint. (a) Moving forward. (b) Moving to the right. (c) Turning.

Static Force Analysis of an Omnidirectional Mobile Robot

349

of the sliding joints. Accordingly, the main body can achieve a three-degree-of-freedom planar motion by driving the wheels at appropriate velocities and can change the moving direction immediately without changing the wheel directions. Thus, SWOM can make an omnidirectional movement. In Fig. 2, the omnidirectional movement of SWOM accompanies the relative movements of the sliding joints. Therefore, SWOM can move within a certain range where the sliders are away from the ends of the rails. On the other hand, when the wheel directions are changed according to the moving direction of the main body as shown in Fig. 3, SWOM can move unlimitedly because the relative movements of the sliding joints do not occur. Then, SWOM should adjust the steering angles of the wheels properly in accordance with the situation.

Fig. 3. Movement of SWOM when the wheel directions are changed depending on the movement direction of the main body. (a) Moving forward. (b) Moving to the right. (c) Turning.

3 Static Force Analysis As explained above, the relative positions of the driving units and the wheel directions vary during the SWOM movement. To evaluate their effect quantitatively, we investigate the characteristics of SWOM focusing on the static force. 3.1 Kinematics and Statics The kinematics of SWOM is explained first. Figure 4 illustrates the reference frames used in the analysis. The global reference frame O − X I Y I is fixed to the floor, while the local reference frame P − X R Y R is fixed to the main body of SWOM. The origin P is centered on the base block, and the axis X R is parallel to one of three sliding joint rails. The position and orientation of the local reference frame in the global frame is  T represented as ξ = x y θ , which corresponds to the pose of the main body. We assign the driving unit attached to the sliding joint parallel to X R as driving unit 1, and the others as driving units 2 and 3 counterclockwise. In the following discussion, the parameters belonging to driving unit i is represented by using the subscript i (i = 1, 2, 3). The angle between the sliding joint rail and X R is αi , the angle between the wheel axle and the rail is βi , the distance between the origin P and the slider is li , and the traveling velocity of the wheel is vi . At this time, the kinematic input-output relation is given by  T Eq. (1), where the input is the velocities of the driving units v = v1 v2 v3 and the output is the velocity of the main body ξ˙ .

350

T. Terakawa and M. Komori

Fig. 4. Reference frames and parameters for SWOM

ξ˙ = Jv

(1)

Here, J is the Jacobian matrix. ⎡

⎤−1 ⎡ ⎤−1 ⎡ ⎤ cos θ sin θ 0 0 − sin α1 cos α1 l1 cos β1 0 J = ⎣ − sin θ cos θ 0 ⎦ ⎣ − sin α2 cos α2 l2 ⎦ ⎣ 0 cos β2 0 ⎦ 0 0 1 − sin α3 cos α3 l3 0 0 cos β3

(2)

The virtual work principle gives the static input-output relation as follows: fd = J T F ⇔ F = J −T fd

(3)

T  where fd = fd 1 fd 2 fd 3 is the input force vector, each of whose element fdi is the driving force of the driving unit i, and F is the output force vector whose direction is the same as that of ξ˙ . When considering the static equilibrium of forces, three kinds of forces, including the driving force fdi , are considered to work on the driving unit as shown in Fig. 5. The lateral force fli is generated due to the contact between the wheel and the ground in the axial direction of the wheel. The lateral force balances with the driving force in the direction parallel to the sliding joint because the passive sliding joint does not transmit any force in the sliding direction. Hence, fli = fdi tan βi .

(4)

The driving unit receives the reaction force fri from the main body via the sliding joint. The direction of fri is perpendicular to the sliding joint rail and the magnitude of

Fig. 5. Forces working on driving unit i

Static Force Analysis of an Omnidirectional Mobile Robot

351

fri equals to that of the resultant force of fdi and fli . fri = −fdi cos βi − fli sin βi = −

fdi cos βi

(5)

3.2 Driving Force Limitation Based on the static equilibrium, we calculate and compare how the maximum force that SWOM can output varies according to the steering angles of the wheels βi and the relative positions of the driving units li . This section discusses the effect of the driving force limitation by assuming that the maximum static friction force is sufficiently larger  T than the driving force. When the output force of the main body is set as F = W 0 0 , i.e., when SOWM puts forth a force in the X I -direction, Eq. (3) yields the following conditions. ⎧ sin(α1 +θ ) sin(α2 +θ) sin(α3 +θ) ⎪ ⎨ W = − cos β1 fd1 − cos β2 fd2 − cos β3 fd3 cos(α2 +θ ) cos(α3 +θ ) 1 +θ ) 0 = − cos(α (6) cos β1 fd1 + cos β2 fd2 + cos β3 fd3 ⎪ ⎩ l1 l2 l3 0 = cos β1 fd1 + cos β2 fd2 + cos β3 fd3 The second and third equations can be summarized as follows: fd 1 fd 2 fd 3 : : = C1 : C2 : C3 (7) cos β1 cos β2 cos β3

where Ci = lj cos(αk + θ) − lk cos αj + θ and (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2). Here, we assume that the driving force of each driving unit is restricted to |fdi | ≤ 1. At this time, the maximum force of the main body is given by    3   i=1 Ci sin(αi + θ ) |W | ≤ . (8) max(|Ci cos βi ||i = 1, 2, 3) With respect to βi , we consider two situations here; one is when the steering angles of the wheels are fixed at βi = 0 as shown in Fig. 2(a) and (b), and the other is when the wheel directions are kept parallel to the moving direction of the main body, i.e., βi = −αi − θ − π/2, like in Fig. 3(a) and (b). Figure 6(a) shows the calculation result of the maximum value of |W | in the case of βi = 0 when θ and l1 are varied under l2 = l3 = 1. The distribution of √ the maximum force for l1 = 1 shapes hexagon. The force reaches the maximum value 3 at θ = nπ/3 and the minimum value 1.5 at θ = (2n − 1)π/6 (n ∈ Z), though one of the driving units does not work at θ = nπ/3 because the output force becomes parallel to the sliding direction of the driving unit. This implies that three driving units can not put forth a large driving force at the same time to keep the momentum equilibrium, which is given by the third equation of Eq. (6). When l1 becomes smaller, the maximum force tend to be large around θ = 0 and π but to be small around θ = π/2 and 3π/2. This is because driving unit 1 has less effect on the momentum equilibrium and can take a large

352

T. Terakawa and M. Komori

driving force. It makes a positive contribution around θ = 0 and π , where driving units 2 and 3 put forth the driving force in the opposite directions, but a negative contribution around θ = π/2 and 3π/2, where driving units 2 and 3 work in the same direction. When l1 becomes larger, the tendency is reversed. At l1 = 2, especially, the output force reaches the global maximum value 2 at θ = π/2 and 3π/2 when all driving units take the maximum driving force. When l1 becomes still larger, however, driving force that driving unit 1 can output is restricted, so that the output force becomes smaller also around θ = π/2 and 3π/2.

1.0 2.0 3.0

(a)

1.0 2.0 3.0

(b)

Fig. 6. Maximum force distributions of SWOM with the driving force limitation under l2 = l3 = 1 when the maximum driving force is set to be 1: (a) βi = 0. (b) βi = −αi − θ − π/2.

Figure 6(b) shows the calculation result of the maximum output force in the case of βi = −αi − θ − π/2 under the same conditions as Fig. 6(a). Compared to Fig. 6(a), the output force becomes larger for each l1 . At l1 = 1, for example, though the minimum value remains 1.5 at θ = (2n − 1)π/6, the output force takes the larger value at the other values of θ and its maximum reaches 2 at θ = nπ/3. When l1 is varied, the distribution of the output force changes almost in the same way as Fig. 6(a), except that the output force becomes large at l1 = 4 in the direction of θ = π/2 and 3π/2. This is because the driving units can utilize the lateral force in addition to the driving force in this situation, whereas driving units 2 and 3 are constrained only by the driving force limitation in Fig. 6(a). At l1 = 4, all driving units can put forth the maximum driving force and the output force takes the global maximum value 3. 3.3 Friction Force Limitation Next, we consider the situation with the friction force limitation, where the resultant force of the driving force and the lateral force is restricted not to exceed the maximum static friction force. Assuming that the center of gravity of SWOM is always on the center of the base block, the momentum equilibrium around the X R - and Y R -axes yields the ratios between the wheel loads of the driving units Ni . N1 : N2 : N3 =

sin(α2 − α3 ) sin(α3 − α1 ) sin(α1 − α2 ) : : l1 l2 l3

(9)

Static Force Analysis of an Omnidirectional Mobile Robot

353

The maximum static friction force f0i is considered to be in proportion to the wheel load Ni . When we set f0i = 1 at l1 = l2 = l3 , its magnitude is given generally as follows: sin(α2 − α3 ) + sin(α3 − α1 ) + sin(α1 − α2 ) lj lk . (10) l2 l3 sin(α2 − α3 ) + l3 l1 sin(α3 − α1 ) + l1 l2 sin(α1 − α2 )  T When SOWM puts forth a force in the X I -direction, i.e., F = W 0 0 , as with the analysis in the previous section, Eq. (3) gives f0i =

Ci cos βi W (11) D where D = l1 sin(α2 − α3 )+l2 sin(α3 − α1 )+l3 sin(α1 − α2 ). Therefore, the condition that the resultant force fri = fdi / cos βi should be smaller than the maximum static friction force f0i concludes    Df0i   . |W | ≤ min  (12) C  fdi =

i

In this case, the maximum value of |W | depends on the relative position of the driving unit li , but not on the steering angle of the wheel βi . Figure 7 shows the calculation result of the maximum force when l1 is changed under l2 = l3 = 1. At l1 = 1, the situation is the same as Fig. 6, so that the maximum force distribution shapes hexagon. The distribution stretches in the direction of θ = π/2, 3π/2 with small l1 , while in the direction of θ = 0 and π with large l1 . This tendency is contrary to those in Fig. 6. This result reflects the distribution of the wheel loads. When the driving unit is closer to the base block, i.e., li is smaller, the wheel load becomes larger, and also the maximum static friction force becomes larger than those of the other driving units. Therefore, when l1 is small, driving unit 1 can put forth a large force, which contributes to a large output in the direction perpendicular to the sliding direction of driving unit 1, namely in the direction of θ = π/2, 3π/2. On the other hand, when l1 is large, driving units 2 and 3 gain the advantage instead of driving unit 1, which realizes the large force at θ = 0 and π . In summary, it is advantageous to make li large when the direction of the output force and the sliding direction of the driving unit is close, whereas to make li small when they are nearly perpendicular.

Fig. 7. Maximum force distribution of SWOM with the friction force limitation under l2 = l3 = 1 when the maximum friction force is set to be 1

354

T. Terakawa and M. Komori

4 Conclusions To solve the problem in conventional mobile robots, the authors proposed SWOM that can perform omnidirectional movement using only normal wheels so as to establish both superb mobility and a simple structure. In this paper, the static force analysis of SWOM was carried out. As a result, it was found that SWOM could gain a large output force under the driving force limitation when the wheels were not steered in the direction perpendicular to the sliding joint and when the relative position of the driving unit whose sliding direction was away from the output direction was properly distant. The result also clarified that it was advantageous to make the distance between the driving unit and the base block large when the direction of the output force and the sliding direction of the driving unit was close considering the friction force limitation.

References 1. Campion, G., Bastin, G.: On adaptive linearizing control of omnidirectional mobile robots. In: Proceedings of the 1989 International Symposium on the Mathematical Theory of Networks and Systems, pp. 531–538. Springer, Heidelberg (1990) 2. Santos, J., Conceição, A., Santos, T., Araújo, H.: Remote control of an omnidirectional mobile robot with time-varying delay and noise attenuation. Mechatronics 52, 7–21 (2018) 3. Chang, H., Wang, S., Sun, P.: Omniwheel Touchdown characteristics and adaptive saturated control for a human support robot. IEEE Access 6, 51174–51186 (2018) 4. Komori, M., Matsuda, K., Terakawa, T., Takeoka, F., Nishihara, H., Ohashi, H.: Active omni wheel capable of active motion in arbitrary direction and omnidirectional vehicle. J. Adv. Mech. Des. Syst. Manuf. 10(6), JAMDSM0086 (2016) 5. Terakawa, T., Komori, M., Yamaguchi, Y., Nishida, Y.: Active omni wheel possessing seamless periphery and omnidirectional vehicle using it. Precis. Eng. 56, 466–475 (2019) 6. West, M., Asada, H.: Design and control of ball wheel omnidirectional vehicles. In: Proceedings of 1995 IEEE International Conference on Robotics and Automation, pp. 1931–1938. IEEE, Piscataway (1995) 7. Ostrovskaya, S., Spiteri, R.J., Angeles, J.: Dynamics of a mobile robot with three ball-wheels. Int. J. Robot. Res. 19(4), 383–393 (2000) 8. Ok, S., Kodama, A., Matsumura, Y., Nakamura, Y.: SO(2) and SO(3), omni-directional personal mobility with link-driven spherical wheels. In: Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 268−273. IEEE, Piscataway (2011) 9. Terakawa, T., Komori, M., Matsuda, K., Mikami, S.: A novel omnidirectional mobile robot with wheels connected by passive sliding joints. IEEE/ASME Trans. Mechatron. 23(4), 1716– 1727 (2018) 10. Terakawa, T., Komori, M., Fujimoto, K.: Control of an omnidirectional mobile robot with wheels connected by passive sliding joints. J. Adv. Mecha. Des. Syst. Manuf. 13(1), JAMDSM0006 (2019)

Development of a Trapezoidal Leaf Spring for a Small and Light Variable Joint Stiffness Mechanism Hiroki Mineshita1(B) , Takuya Otani2,3 , Kenji Hashimoto3,4 , Masanori Sakaguchi5 , Yasuo Kawakami6 , Hun-ok Lim3,7 , and Atsuo Takanishi3,8 1 Graduate School of Creative Science and Engineering, Waseda University, Shinjuku, Tokyo,

Japan [email protected] 2 Research Institute for Science and Engineering, Waseda University, Shinjuku, Tokyo, Japan 3 Humanoid Robotics Institute, Waseda University, Shinjuku, Tokyo, Japan [email protected] 4 School of Science and Technology, Meiji University, Kawasaki, Kanagawa, Japan 5 Institute of Sports Science, ASICS Corporation, Kobe, Hyogo, Japan 6 Faculty of Sports Science, Waseda University, Tokorozawa, Saitama, Japan 7 Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan 8 Department of Modern Mechanical Engineering, Waseda University, Shinjuku, Tokyo, Japan

Abstract. Herein, we have developed a humanoid robot that achieves dynamic motion. Focusing on the running motion that is the basis of the motion, the robot has been developed focusing on the pelvic rotation on the frontal plane and the elasticity in leg joints (that changes according to running speed), which are the characteristics of humans during running. However, the variable joint stiffness mechanism that we have developed was large and heavy. Therefore, to make the mechanism smaller and lighter, we shorten the length of the leaf spring. We succeeded in downsizing the mechanism by changing its rectangular shape to trapezoidal, while maintaining strength and elasticity. The variable joint stiffness mechanism thus developed was more flexible, and its weight was reduced from 1.9 kg to 0.7 kg. The mechanism was mounted on the ankle joint, and it was confirmed that the required specifications were satisfied. Keywords: Joint stiffness mechanism · Series elastic actuator · Leaf spring · Humanoid · Running robot

1 Introduction Several biped humanoid robots have realized dynamic motions. Honda’s ASIMO [1] runs at 9 km/h, and Boston Dynamics’ Atlas [2] runs at 5.4 km/h, as well as performs somersaults and parkour. Although these robots have joints like humans, they are driven only by electric and hydraulic power, and do not have the elasticity like that in the human joints. Humans use the elasticity in the leg joints when running [3] or jumping [4], and © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 355–363, 2021. https://doi.org/10.1007/978-3-030-58380-4_43

356

H. Mineshita et al.

Fig. 1. WATHLETE-1. The robot has 22 DOFs. Blue, red, and yellow joints denotes the roll, pitch, and yaw axes, respectively, in (b).

achieve efficient dynamic motions. While running, humans attain efficiency by changing the joint stiffness according to the running speed [5]. We have developed the biped humanoid robot, WATHLETE-1 (Waseda ATHLETE humanoid No. 1; Fig. 1), focusing on the elasticity in the leg and the rotation observed in the pelvis during running. This robot has the same link length and mass as humans, has elastic elements composed of leaf springs made of carbon fiber reinforced plastic (CFRP) at the knee and ankle joints, and has a variable joint stiffness mechanism at the knee joint [6]. However, the weight of current variable joint stiffness mechanism is 1.9 kg, and the structure is large. This is because the leaf spring becomes large to withstand the torque applied during running, and the mechanism becomes large accordingly. In addition, no available joints can be satisfactorily driven during dynamic motions such as running; while running, fine control is not possible as the motion at the joints are limited to passive rotations of the leg. Therefore, reducing the size and weight of the variable joint stiffness mechanism would help mount a high output motor and improve the output of the joints. To reduce the size of the mechanism, it is necessary to reduce the size of the leaf spring. However, it is difficult to reduce the size of the rectangular leaf spring because of strength and elasticity considerations. Therefore, we concluded based on our analysis that the size can be changing the shape of the leaf spring.

2 Design and Mechanism 2.1 Techniques for Shortening Leaf Springs With respect to the shape of the leaf spring, we considered the necessary requirements to maintain strength and reduce stiffness. First, regarding the strength, the maximum stress is generated at the fixed point of the leaf spring. On the other hand, lowering stiffness leads to low strength. Therefore, sufficient strength could be secured at the fixed point where the maximum stress was applied, and the strength could be reduced at the tip. Therefore, alternation of the width was considered, and a trapezoidal shape was adopted. The analysis of trapezoidal leaf springs is based on the expansion method. This method is defined in Japanese Industrial Standards (JIS) B 2710, and is an approximation that is used for multi-leaf spring calculations (Fig. 2(a)). This multi-leaf spring is considered as same as the leaf spring expanded toward the width direction on the same plane. When

Development of a Trapezoidal Leaf Spring

357

the thickness of each leaf spring is the same, a trapezoidal shape can be approximated (See Fig. 2(b)).

Fig. 2. (a) Multi-leaf spring subject to the expansion method. (b) Approximation model by expansion model. In (a) and (b), n is twice the number of leaf springs, n is twice the number of leaf springs with the total length of 2l n , and b is the width of each leaf spring.

This trapezoidal leaf spring can be calculated as follows: First, calculate the coefficient K T based on the trapezoidal shape using 

KT =

n  η = n , and2  3  − 2η + η 2 − loge η .

3 1 (1−η)3 2

(1)

The spring coefficient (k) of the trapezoidal leaf spring can be calculated using K T . k=

1 nEbt 3 KT 4l 3

(2)

where E is the Young’s modulus and t is the thickness of the leaf spring. Furthermore, the maximum stress (σ ) at the fixed point can be calculated as σ =

6M nbt 2

(3)

where M is the moment applied to the leaf spring. As the aforementioned formula is calculated for a multi-leaf spring, the calculation of a trapezoidal leaf spring should have nb as the width of the root and n b as the width of the tip. Similarly, applying the expansion method to the calculation of the laminated leaf springs of a trapezoidal leaf spring leads to an equation that simply replaces b with the root width in Eq. (2) and (3). The calculations confirmed that the same stiffness can be achieved even if the length is shorter than the rectangular shape. This reduces the leaf spring while maintaining strength. In this study, the stiffness required for elasticity at the joints is low, and the joints deforms considerably. A rotation of 33° will be observed at the mounted ankle joint while running at 5 m/s. When such deformations are involved, the assumption of small deformations in the above equations breaks down, and errors occur in the calculations. Therefore, we have used equations that do not use approximation, so that they can handle large deformation [7, 8]. By repeatedly calculating the equations, it became possible to calculate the change in stiffness owing to the displacement between the rotational center and the fixed point of the leaf spring [6] and the displacement of the load point caused by

358

H. Mineshita et al.

the deformation of the leaf spring. In this method, the curve along the deflection of the leaf spring is defined as the s-axis, and the deflection angle ϕ(s) is defined and calculated based on the s-axis. At that time, ϕ(s) is calculated EI (s)

d ϕ(s) =M ds

(4)

where I(s) is the moment of inertia of the area, which varies with the cross-sectional shape. As the trapezoidal shape is used this time, the cross-sectional shape changes along the s-axis. The effect of this change corresponds to the shape factor (K T ) in the trapezoidal calculation. Calculations using this equation show that the trapezoidal shape makes the elasticity nonlinear because the cross-sectional shape changes owing to the shift of the load point. However, this can be counteracted by using the effect of the displacement between the rotational center and the fixed point of the leaf spring. In this study, a variable joint stiffness mechanism is mounted on the ankle joint using the trapezoidal leaf spring. 2.2 Design of the Leaf Spring in Ankle Variable Joint Stiffness Mechanism The required specifications for the ankle stiffness are listed in Table 1, and the stiffness is low with respect to the joint torque. This was calculated from anthropometric data [9]. Therefore, it is difficult to achieve stiffness in one place. To solve the aforementioned problem, we have connected the leaf springs in series and decided to disperse them at the foot and lower legs. Foot stiffness is constant and designed to have a joint stiffness of 400 Nm/rad. The above calculation without approximation showed that the stiffness value changed depending on the torque (Fig. 4(a)). The ankle joint is driven by the 4-bar linkage of the lower leg (Fig. 3), but the reduction ratio is non-linear (Fig. 4(b)). To counteract this effect, we decided to design the lower leg leaf spring so that its stiffness decreased slightly as the torque increased. The effective length was 110 mm owing to the mounting space of the lower leg. From these conditions, as shown in Fig. 5, it is possible to design by using the equations that do not use approximation, so that the stiffness does not change with respect to the torque at the joint (Fig. 6). If a leaf spring of the same specification is developed rectangularly, it would be 30 mm longer. Thus, it is necessary to increase the effective length of the variable joint stiffness mechanism from 20 to 100 mm. Table 1. Torque and quasi-stiffness of ankle joint while running Running Speed m/s 2.0 3.5 5.0

Torque Nm 136 178 190

Human Quasi-Stiffness Nm/rad 250 305 325

Previous Mechanism Torque Quasi-Stiffness Nm Nm/rad 65

318

Development of a Trapezoidal Leaf Spring

359

Fig. 3. Ankle mechanism

Fig. 4. Parameters affecting the design of the lower leg spring. (a) Relationship between the torque and stiffness of the foot spring. (b) Relationship between reduction ratio and joint angle (negative denotes plantar flexion, and positive denotes dorsiflexion)

Fig. 5. Design parameters of the lower leg leaf spring

2.3 Design of the Ankle Variable Joint Stiffness Mechanism Requirements for the design of the variable joint stiffness mechanism are as follows: The position of the load point can be kept constant during standing phase while running, the stiffness is increased once to suppress the vibration of the leg during swing phase, and the load point is moved to the target stiffness value by changing effective length.

360

H. Mineshita et al.

Fig. 6. Changes in the relationship between the torque and the stiffness considering the effective length (a) at the lower leg spring and (b) at the ankle joint

The reaction force is applied to the load point of the leaf spring, and its direction changes owing to the deflection of the leaf spring. Of these, the load which is the same direction as the moving of the load point must be withstand by the actuator. The force is ~600 N. The load point needs to reciprocate up to 90 mm during the swing phase (0.4 s). The thrust required to move the load point is calculated from inertia and friction to be up to 10 N. Until now, brakes have been used to withstand forces during the stance phase, but has been heavy. Therefore, we have examined worms and sliding screws with the selflocking function; however, the speed requirements could not be satisfied. Therefore, we have tried to reduce the load on the motor caused by the reverse transmission efficiency. To meet the requirements, a small motor is used with a reduction ratio of 10 and 75% reverse efficiency. On the other hand, because of the reaction force from the leaf spring, the load applied perpendicularly to the moving direction reaches 9000 N, which is the maximum. Therefore, it was decided to withstand the load by the LM roller, its rated load is 15000 N despite its small size and to use liner guides to receive other loads. In the past, a plate was added to fix the linear guides of the variable joint stiffness mechanism. However, in this study, the plate was placed outside the link to secure space. Thus, the weight was reduced from 1.9 to 0.7 kg (See Fig. 7).

Fig. 7. Comparison between previous and new variable stiffness mechanisms

3 Verification 3.1 Verification of Laminated Trapezoidal Leaf Spring First, the developed trapezoidal leaf spring using in lower leg has been verified using a conventional load experimental machine [6] (See Fig. 8(a)). The actual value was

Development of a Trapezoidal Leaf Spring

361

compared with the theoretical values calculated using no approximation and using JIS formula. The experimental result is shown in Fig. 8(b). The difference between measurement and theory is because the deformation on the experimental machine is not considered. Also, the value calculated using JIS formula is closer to the experimental value. This is because the approximation of JIS formula calculates more deflection than it actually does.

Fig. 8. Experimental environment and result. Load is applied to the leaf spring with a universal testing machine (a). (b) The deflection results when the effective length is 100 mm.

3.2 Evaluation of Stiffness Variable Range and Its Change Owing to Torque The developed variable joint stiffness mechanism is evaluated. In this experiment, the stiffness has been changed manually to evaluate the stiffness of the mechanism. Figure 9 shows the change in elasticity caused by torque at each effective length. When the load is less, the stiffness decreases owing to the existence of the gap between the leaf springs, but when the effective length of the foot is 100 mm, which is same as the design, there is little change in stiffness owing to torque (See Fig. 9(a)). However, the joint stiffness is lower than the required specification. This is because the stiffness calculated is high in the equations that do not use approximation, which has been found in the aforementioned experiment. However, the required specification can be met by reducing the effective length of the foot from 100 to 30 mm. In this case, the stiffness changes significantly due to the torque. Based on this, the relationship between torque and stiffness can be adjusted using detailed calculations. Improving the accuracy of the equations without approximation can form a topic for future study. And in the future, we will assemble the mechanism so that the joint stiffness can be changed automatically.

362

H. Mineshita et al.

Fig. 9. Relationship between ankle joint quasi stiffness and torque at each lower leg spring’s effective length. Green area in (b) is required specification. In (a), Requirement is outside the graph.

4 Conclusion In this study, the trapezoidal leaf spring was designed for a small and lightweight variable joint stiffness mechanism. The mechanism was designed and mounted on the ankle joint mechanism. The weight of the variable joint stiffness mechanism was significantly reduced, and the required specifications could be realized by adjusting the effective length. Furthermore, this mechanism was designed to keep the stiffness constant when the torque changed, considering the effects of large deformations. In the future, we will focus on modifying the theoretical equations and distributing the mechanical elements of the legs to reduce the weight of the ankle mechanism. Acknowledgements. Research was conducted with the support of Research Institute for Science and Engineering, Waseda University; Humanoid Robotics Institute, Waseda University; Human Performance Laboratory, Waseda University; Future Robotics Organization, Waseda University. It was also financially supported in part by NSK Foundation for the Advancement of Mechatronics and the JSPS KAKENHI Grant No. 17H00767. Further, 3DCAD software SolidWorks was provided by SolidWorks Japan K.K.; cables and connectors were provided by DYDEN CORPORATION.

References 1. Honda Global | ASIMO. https://global.honda/innovation/robotics/ASIMO.html#2011. Accessed 11 Feb 2020 2. Atlas® | Boston Dynamics. https://www.bostondynamics.com/atlas. Accessed 21 Nov 2019 3. McMahon, T.A., Cheng, G.C.: The mechanics of running: How does stiffness couple with speed? J. Biomech. 23, 65–78 (1990) 4. Stefanyshyn, D.J., Nigg, B.M.: Contribution of the lower extremity joints to mechanical energy in running vertical jumps and running long jumps. J. Sports Sci. 16(2), 177–186 (1998) 5. Arampatzis, A., Brüggemann, G.-P., Metzler, V.: The effect of speed on leg stiffness and joint kinetics in human running. J. Biomech. 32(12), 1349–1353 (1999) 6. Otani, T., et al.: Joint mechanism that mimics elastic characteristics in human running. Machines 4(1), 5 (2016)

Development of a Trapezoidal Leaf Spring

363

7. Bisshopp, K.E., Drucker, D.C.: Large deflection of cantilever beams. Quart. Appl. Math. 3(3), 272–275 (1945) 8. Mutyalarao, M., Bharathi, D., Rao, B.N.: Large deflections of a cantilever beam under an inclined end load. Appl. Math. Comput. 217(7), 3607–3613 (2010) 9. Otani, T., et al.: Utilization of human-like pelvic rotation for running robot. Front. Robot. AI 2, Article no. 17 (2015)

Connecting MATLAB/Octave to Perceptual, Cognitive and Control Components for the Development of Intelligent Robotic Systems Enrique Coronado(B) , Liz Rincon, and Gentiane Venture Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-0012, Japan [email protected]

Abstract. This article presents a novel approach for interfacing MATLAB and Octave programs with Robotics and Internet of Things (IoT) software modules. The proposed library, developed in Java, is part of NEP, an open and cross-platform distributed robotic framework. This library is a high-performance and cross-platform alternative that enables developers to build more complex cognitive and control architectures with MATLAB and Octave. For evaluation, the latency for different message sizes against options using the ROS toolbox and rosbridge suite is compared. Minimal code samples and examples of research applications connecting perceptual and cognitive modules for the creation of adaptive and emotional intelligent robots are also presented. Keywords: Robot modules robots

1

· Human-robot interaction · Adaptive

Introduction

Nowadays many robotics and control researchers still consider the use of highlevel numerical computing environments, such as MATLAB and Octave, as keystone software tools enabling prototyping, simulation, and validation of mechanisms, robot algorithms, and control systems. In order to create more advanced robotics systems, programs written in MATLAB or Octave must be able to interface with sensory, perceptual, and cognitive modules in the localhost (i.e., inside the same computer) and remotely (i.e., between different computers) [16]. These modules often require their execution in different programming languages. A common way to enable the connection between two isolated processes is the use of conventional and synchronous TCP/IP BSD sockets [14,21]. However, the use of conventional BSD sockets limits developers to the creation of Client/Server architectures [1]. Compared with the Publish/Subscribe pattern, a Client/Server solution tends to have lower streaming data performance and creates less reliable software architectures [9,16]. Some alternatives to conventional BSD sockets c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 364–371, 2021. https://doi.org/10.1007/978-3-030-58380-4_44

Connecting MATLAB/Octave to Perceptual, Cognitive and Robot Control

365

enabling the interfacing of MATLAB with external robotics modules have been proposed in the literature recently [7,13,22,23]. Recently, the Robot Operating System (ROS) and Robotics System Toolboxes [7] were launched to enable the creation of ROS components in MATLAB. However, programs applying this approach can only communicate with other modules using the same version of ROS. A popular option used to connect different versions of ROS (i.e., ROS 1.0 with ROS 2.0) is the rosbridge suite [11]. This implementation of the rosbridge protocol has been used in many projects to connect programs using ROS with modules using programming languages or executed from operating systems not supported by ROS. On the other hand, BSD sockets alternatives supporting Octave are rarely reported in the literature. An exception is rosoct [2], which has not been supported since 2010. Therefore, recent versions of Octave (from 3.0 and above) are not compatible with this library. Unlike MATLAB, Octave is open source and free to use. The availability of inter-process communication methods that enable the creation of distributed robotic systems in open-source platforms, such as Octave, is, in fact, a relevant contribution to the community. This can be especially valuable for those researchers, universities, students, and practitioners using Octave due to their personal preferences or license limitations (which makes the use of MATLAB a non-possible option). Therefore, the main contribution of this article is the development of a Javabased NEP library to enable MATLAB/Octave programs to communicate with robotic and Internet-of-Things (IoT) modules using high-performance sockets. This connection can be performed using either the Client/Server or Publish/Subscribe patterns. NEP for MATLAB/Octave is built over ZeroMQ, which enables high-performance data streaming with software modules written in different languages and executed in most desktop and mobile operating systems [1]. In this work, a sample code for creating Publish/Subscribe communication and examples of cognitive architectures using NEP in MATLAB are presented. Suitability of the proposed approach is evaluated by comparing latency over a remote network with two state-of-the-art methods connecting ROS 1.0 and ROS 2.0 modules. Moreover, latency comparison with the official rosbridge suite connecting MATLAB with non-ROS enabled programming language (Javascript over Node.js) is also presented. Unlike solutions based on the rosbridge protocol, and the ROS toolbox for MATLAB, our approach is ROS-independent. NEP can be used for both enabling the creation of architectures requiring or not the integration of ROS frameworks. Moreover, NEP for MATLAB/Octave can be use as a high-performance alternative to rosbridge enabling communication between ROS-based modules (either ROS 1.0 or ROS 2.0) as well as non-ROS enable software components with MATLAB or Octave. Finally, we summarize novel cognitive architectures using NEP for enabling research in emotional intelligent social robots and adaptive control systems.

2

NEP for MATLAB and Octave

NEP is a novel robot communication framework for building distributed software architectures for robotics systems. The initial and deprecated prototype of NEP,

366

E. Coronado et al.

described in [8], only supported Python 2. The current version of NEP [10] supports Python 2, Python 3, C#, and JavaScript. NEP is built on top of low-level socket communication libraries and robotics middlewares, which can be selected as back-end options. The main contribution of this work is a MATLAB and Octave compatible version of NEP written in Java and based in ZeroMQ sockets, which is the default back-end option of NEP. Middleware solution based on ZeroMQ can be designed to be lightweight and portable without compromising performance. Examples of non-robotics communication frameworks and middle-wares build on top of ZeroMQ are presented in [4,12,18]. Similar to ROS 2.0, which is built on top of Data Distribution Services (DDS) and YARP [17], which is built on top of the Adaptive Communication Environment (ACE) library [6,20], NEP adapts and simplifies the use to a low-level and general purpose communication framework, and offers a set of supporting software tools for enabling the creation of advanced robot and IoT software architectures. NEP for MATLAB and Octave supports both Client/Server and Publish/Subscribe communication patterns. As an example, the minimal code required to create a publisher node in NEP using MATLAB is shown below: 1 2 3 4 5 6 7 8 9 10

import nep.Node; % use nep library node = nep.Node("matlab_sender"); % create a new nep node pub = node.new_pub("test", "json"); % Define publisher for c = 1:5 % Publish 5 times msg = struct(’message’,c) % Messages defined as structures json_msg = jsonencode(msg) % Convert matlab structures to JSON pub.publish(json_msg) pause(1) end pub.close() % Close subscriber

NEP can use a ROS-like hybrid peer-to-peer network architecture management and configuration (more details in [15]). Therefore, the node name that identifies the software module (line 2), and the topic used to filter the messages sent (line 3) must be defined. In the above example, the topic is denoted as test. NEP messages are denoted with JavaScript Object Notation (JSON). These messages can be easily encoded and decoded to MATLAB structures using JSON serializable data types. Minimal code required to create a subscriber able to read information from the publisher defined above is shown below: 1 2 3 4 5 6 7 8 9 10 11

import nep.Node; % use nep library node = nep.Node("matlab_receiver"); % create a new nep node sub = node.new_sub("test","json"); while 1 % read data msg = sub.listen(); % listen data if strcmp(msg,"{}") % If message is null pause(.001) else %Covert JSON to matlab structure value = jsondecode(string(msg)) end end

A similar approach can be applied to generate a Publish/Subscribe architecture in Octave. The listen() function can be defined in non blocking mode; therefore, enabling developers the use multiple subscribers in a node. These and more sample examples enabling Client/Server and Publish/Subscribe architectures in both MATLAB and Octave as well as instructions for installing NEP tools and libraries are available in [3].

Connecting MATLAB/Octave to Perceptual, Cognitive and Robot Control

367

Fig. 1. Latency comparison connecting MATLAB/Octave with Python using ROS Toolbox and NEP

3

Evaluations

In order to prove the technological suitability of NEP for MATLAB/Octave, we compare its communication performance against relevant state-of-the-art solutions. For this task, the round-trip latency (time from the publisher in MATLAB/Octave to the destination in other programming language plus the time from the destination back to MATLAB/Octave) is measured between: a) MATLAB and Python using ROS Toolbox, b) MATLAB and Python using NEP and c) OCTAVE and Python using NEP. It is relevant to highlight that the ROS toolbox is not compatible with Octave. These options simulate cases when control programs written in MATLAB/Octave require to be connected with external software modules written in different programming languages and executed in a remote computer. In these evaluations, MATLAB 2019b and Octave 5.1.0 are executed in a Surface PRO 4 with an Intel Core i5-8350U with 8 GB of RAM and Windows 10. Remote PC running the external programs interfacing with MATLAB are executed in a Laptop Dell Inspiron 14 with an Intel Core i7-7500U, 8 GB of RAM, and Ubuntu 16. Wi-Fi connection is performed using a portable TPlink TL-WR802N modem for connecting these two computers. Figure 1 shows the latency results for different message sizes executing the proposed scenarios. These values represent the mean of 200 samples for each message size. Results indicate that b) NEP in MATLAB highly outperforms latency results of a) ROS Toolbox in MATLAB and c) NEP in Octave. To prove the suitability of using NEP for interfacing MATLAB/Octave with non-ROS enable software modules round-trip latency is registered between a) MATLAB and node.js using ROS Toolbox and the rosbridge suite, b) MATLAB and node.js using NEP and c) OCTAVE and node.js using NEP. Results for different message sizes are shown in Fig. 2. Results for communicating with Node.js also show lower latency using NEP and MATLAB. However, in this case the highest latency is reached when using rosbridge to communicate with Node.js. This can be due to the fact that messages must pass through a rosbridge server, which converts messages from ROS sockets to Websockets. Instead NEP enables direct communication between MAT-

368

E. Coronado et al.

Fig. 2. Latency comparison connecting MATLAB/Octave with Node.js using rosbridge and NEP

LAB/Octave and external modules written in many other programming languages. Moreover, an improvement in latency when interfacing with Node.js is also obtained. This is an expected result as Node.js is built on top of C++, which is known to be of higher performance than Python.

4

Example of Applications

This section describes a general system architecture, which uses NEP to connect robot components performing perception, cognition, and robot control. This architecture is involved in our research on the development of social, intelligent, and adaptive robots. The architectures are applied using MATLAB for optimal predictive, and robust motion-controllers. The interaction between these modules is summarized in Fig. 3. All these modules, which are written in different programming languages, are connected using NEP. A deep description of these applications are shown in [5,19]. The main parts of the architecture includes: i) A set of data acquisition modules obtaining sensory information from the real-world; ii) Perceptual modules, which subscribe and transform the low-level sensory data into high-level representations (e.g, gestures, human actions, behaviors, and emotions); iii) A module that manages decision-making using high-level knowledge; iv) The robot controller (written in MATLAB), which continuously receives information from sensory, perceptual and decision-making modules thought different NEP topics; and v) a web-based user interface displaying sensory, perception, and cognitive states. We use continuous and dimensional values of emotions to adapt parameters of robot controllers (enabling more emotional reactions and expressions). An example of an application using NEP and MATLAB for the development of an emotionally intelligent robot is described below. In this application, the main controllers and cognitive models are executed in MATLAB. These modules communicate with sensorial/perceptual components, which are written in Python 2 and 3. Mainly, the robots are working with predictive, optimal, and robust controllers, in adaptive modality, changing continuously information from

Connecting MATLAB/Octave to Perceptual, Cognitive and Robot Control

369

Fig. 3. General cognitive and adaptive architecture for emotional robots

the robot perception which is communicated by NEP to adapt their values. The control data is sent as a predefined sample time according to the robot system. The performance of different kinds of controllers programmed in the same platform (hardware and software) depends directly on the design of each controller, and it can differ depending on the control structure. The performance of the NEP-MATLAB is evidenced when the data is applied to real-time changing controllers and getting the robot actions. The latency can influence the time delay response in the robot motions. As Fig. 2 showed the latency, if the system is implemented in different software platforms (NEP-Octave or NEP-ROS) with the same kind of controllers, it can react in different time, due to the latency produced according to the data streaming in the nodes. The robot motions are influenced in the final time response, however, the robot controllers are created in a stable and robust regions involving disturbances, time delay in the inputs, noise in the signal and variation of the dynamic parameters. The system performance can be improved with different hardware platforms with more capabilities for data acquisition, processing and network communication. One of our applications was the development of an interactive robot able to interact and react according to the human engagement and intentions while executing priority tasks [19]. The application uses adaptive controllers and deep perceptual models (Fig. 4a). It is applied to a 5DoF robot using Adaptive Generalized Predictive Controllers developed in MATLAB. It is integrated with human motion and emotion recognition. Perception is developed in Python using deep learning models for face detection by Multi-task cascaded CNN (Convolutional Neural Networks), emotion detection using Microsoft Face API with a K-Nearest Neighbors classifier, human engagement by proximity sensing, and hand gesture, all of these modules running in Python 3 and using NEP for communication. Another application is adaptive and predictive controllers for expressive robot arm movements during the human and environment interaction [5]. It was developed to design more effective robot motions with non-anthropomorphic characteristics. The perception is generated by the human, the environment and the overall interaction (Fig. 4b). It is configured in Python and manages the sensory data by NEP. For cognition and decision making, Adaptive Fuzzy Emotional Models are developed. The robot motion and control use MATLAB in realtime. Robust Generalized Predictive Controllers are developed, which applied

370

E. Coronado et al.

(a)

(b)

Fig. 4. Expressive and cognitive robots with adaptive predictive controllers using NEP with Python and MATLAB programs for fast perception and optimal motion control

optimization cycles to perform stable robot actions. The feedback loops are executing in real-time via NEP in continuous flow.

5

Conclusions

This article presented the initial release of NEP for MATLAB and Octave. Evaluation results show that: i) NEP in MATLAB outperforms latency performance reached by ROS Toolbox; and ii) NEP in Octave presents acceptable latency performance, which is similar to that of ROS Toolbox. This article also demonstrates the suitability of NEP for connecting recent MATLAB/Octave versions with other programming languages (i.e. Python and Node.js) over a remote network. Therefore, the presented results are highly dependent on the robustness and performance of the WiFi connection. In the case of this article a low-cost and portable WiFi modem is used. Moreover, presented applications prove the usability of NEP for enabling the development of novel cognitive architectures using MATLAB or Octave. Rather than be designed to substitute ROS and the rosbridge suite, researchers can use NEP libraries to extend existing software architecture using or not ROS to interface with external perceptual, cognitive, or control modules, which can be written in different programming languages and executed in different operating systems or smart-devices.

References 1. 2. 3. 4.

https://zeromq.org/ (2019) http://wiki.ros.org/rosoct (2019) https://enriquecoronadozu.github.io/NEP/ (2019) Al-Turany, M., Buncic, P., Hristov, P., Kollegger, T., Kouzinopoulos, C., Lebedev, A., Lindenstruth, V., Manafov, A., Richter, M., Rybalchenko, A., et al.: Alfa: the new alice-fair software framework. In: Journal of Physics: Conference Series, vol. 664, p. 072001. IOP Publishing (2015) 5. Ardila, L.R., Coronado, E., Hendra, H., Phan, J., Zainalkefli, Z., Venture, G.: Adaptive fuzzy and predictive controllers for expressive robot arm movement during human and environment interaction. Int. J. Mech. Eng. Robot. Res. 8(2), 207–219 (2019)

Connecting MATLAB/Octave to Perceptual, Cognitive and Robot Control

371

6. Chitic, S.G., Ponge, J., Simonin, O.: Are middlewares ready for multi-robots systems? In: International Conference on Simulation, Modeling, and Programming for Autonomous Robots, pp. 279–290. Springer (2014) 7. Corke, P.: Integrating ROS and MATLAB [ROS topics]. IEEE Robot. Autom. Mag. 22(2), 18–20 (2015) 8. Coronado, E., Mastrogiovanni, F., Venture, G.: Design of a human-centered robot framework for end-user programming and applications. In: ROMANSY 22–Robot Design, Dynamics and Control, pp. 450–457. Springer (2018) 9. Coronado, E., Mastrogiovanni, F., Venture, G.: Development of intelligent behaviors for social robots via user-friendly and modular programming tools. In: 2018 EEE International Workshop on Advanced Robotics and its Social Impacts (ARSO). IEEE (2018) 10. Coronado, E., Venture, G.: Towards IoT-aided human-robot interaction using NEP and ROS: a platform-independent, accessible and distributed approach. Sensors 20(5), 1500 (2020) 11. Crick, C., Jay, G., Osentoski, S., Pitzer, B., Jenkins, O.C.: Rosbridge: ROS for non-ROS users. In: Robotics Research, pp. 493–504. Springer (2017) 12. Dworak, A., Ehm, F., Charrue, P., Sliwinski, W.: The new cern controls middleware. In: Journal of Physics: Conference Series, vol. 396, p. 012017. IOP Publishing (2012) 13. Hold-Geoffroy, Y., Gardner, M.A., Gagn´e, C., Latulippe, M., Giguere, P.: ros4mat: A matlab programming interface for remote operations of ROS-based robotic devices in an educational context. In: 2013 International Conference on Computer and Robot Vision, pp. 242–248. IEEE (2013) 14. Jones, M.T.: BSD Sockets Programming from a Multi-Language Perspective. Charles River Media, Inc., Newton (2003) 15. Joseph, L., Cacace, J.: Mastering ROS for Robotics Programming: Design, Build, and Simulate Complex Robots Using the Robot Operating System. Packt Publishing Ltd., Birmingham (2018) 16. Kortenkamp, D., Simmons, R., Brugali, D.: Robotic systems architectures and programming. In: Springer Handbook of Robotics, pp. 283–306. Springer (2016) 17. Metta, G., Fitzpatrick, P., Natale, L.: Yarp: yet another robot platform. Int. J. Adv. Rob. Syst. 3(1), 8 (2006) 18. Mirabito, L.: Zdaq, a light data acquisition framework based on zeromq. J. Instr. 14(10), C10,007 (2019) 19. Rincon, L., Fillol, F., Coronado, E., Venture, G.: Adaptive optimal predictive control system for cognitive manipulator robots based on human engagement/intention and deep dynamic perception. In: Proceedings of the 25th Jc-IFToMM Symposium, pp. 23–30. IFToMM (2019) 20. Schmidt, D.C.: The adaptive communication environment: an object-oriented network programming toolkit for developing communication software (1993) 21. Sechrest, S.: An introductory 4.4 BSD interprocess communication tutorial. Computer Science Research Group, Department of Electrical Engineering and Computer Science, University of California, Berkeley (1986) R version 0.8: an updated communication system 22. Wason, J.D.: Robot raconteur for robotics, automation, building control, and the internet of things. In: 2016 IEEE International Conference on Automation Science and Engineering (CASE), pp. 595–602. IEEE (2016) 23. Yang, C., Ma, H., Fu, M.: Human–robot interaction interface. In: Advanced Technologies in Modern Robotic Applications, pp. 257–301. Springer (2016)

Balancing of Planar 5R Symmetrical Parallel Manipulators Taking into Account the Varying Payload Jing Geng1,2(B) and Vigen Arakelian1,2(B) 1 Mecaproce/INSA-Rennes, 20 av. des Buttes de Coesmes, CS 70839, 35708 Rennes, France

{jing.geng,vigen.arakelyan}@insa-rennes.fr 2 LS2N-ECN UMR 6004, 1 rue de la Noë, BP 92101, 44321 Nantes, France

Abstract. The shaking force balancing are carried out by an optimal redistribution of moving masses, which allows one to cancel or reduce the variable dynamic loads on the manipulator frame. The shaking force balancing mostly leads to an increase in the moving mass of the manipulator, which has negative impact on the input torques and the shaking moment. It was shown that the force balancing method based on the minimization of dynamic loads on the manipulator frame via reducing the acceleration of the common centre of mass of moving links is more optimal from this point of view. To minimize the acceleration of the manipulator’s common centre of mass, and subsequently, the shaking force, the “bang-bang” law is used. In this paper, it is shown that such a solution is also efficient when the manipulator must be balanced taking into account the varying payload. Numerical simulations illustrate the simplicity and the efficiency of the suggested balancing technique. Keywords: Force balancing · Shaking force · Payload · Parallel manipulator · 5R parallel mechanism · “bang-bang” law

1 Introduction One of the important design stages in the development of high-speed manipulators is the shaking force balancing. The aim of the shaking force balancing is the reduction of variable dynamic loads on the manipulator frame, which are sources of vibrations. Different design concepts devoted to the shaking force balancing of manipulators have been developed and documented [1]. However, basically, the methods of shaking force balancing are achieved by an unavoidable increase of the total mass of moving links. The force balancing can be reached by adding counterweights [2–5] or auxiliary structures [6, 7]. The most important deficiency of these solutions is the increase in the total mass of moving links. Therefore, in the study [8] has been developed a balancing approach based on the optimal trajectory planning of the common centre of mass of the manipulator. The aim of the developed balancing method consists in the fact that the manipulator is controlled not by applying end-effector trajectories but by planning the displacements of the total mass centre of moving links. The trajectories of the total mass centre of © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 372–379, 2021. https://doi.org/10.1007/978-3-030-58380-4_45

Balancing of Planar 5R Symmetrical Parallel Manipulators

373

the manipulator are sated as a straight line between the initial and final positions of the end-effector. Then, the motion between these positions is parameterized with “bangbang” motion profile. It allows the reduction of the maximal value of the centre of mass acceleration and, consequently, leads to the reduction in the shaking force. This method found further development in [9–15]. In the present study, the authors propose to apply the mentioned balancing method on 5R parallel manipulators, taking into account the varying payload. Note that during such balancing, the counterweights are not added to the manipulator links, and the reduction of inertial forces is carried out by optimal trajectory and acceleration generation of the common centre of masses of the 5R parallel manipulator.

2 Shaking Force Balancing Taking into Account the Payload 2.1 Problem Formulation A kinematic scheme of the planar 5R parallel manipulator is shown in Fig. 1. The output axis P(x, y), which corresponds to the axis of the end-effector, is connected to the base by two legs, each of which consists of three revolute joints and two links. The two legs are connected to a common axis P with the common revolute joint at the end of each leg. In each of the two legs, the revolute joint connected to the base is actuated. Such a manipulator can position the end-effector freely in a plane. In the given planar 5R symmetrical parallel mechanism each actuated joint is denoted as Ai (i = 1, 2), the other end of each actuated link is denoted as Bi (i = 1, 2) and the common joint of the two legs is denoted as P, which is also the axis of the end-effector. A fixed global reference system Oxy is located at the centre of A1 A2 with the y-axis normal to A1 A2 and the x-axis directed along A1 A2 . The lengths of links are denoted as l1 = A1 B1 = A2 B2 , l2 = B1 P = B2 P and l0 = OA1 = OA2 . The locations of the centres of mass are denoted as r1 = lA1 S1 = lA2 S2 and r2 = lB1 S3 = lB2 S4 . The mass of link 1, 2 is m1 and the mass of link 3, 4 is m2 .

Fig. 1. The planar 5R parallel manipulator.

The shaking forces Fsh of the 5R symmetrical parallel manipulator with payload can be written as:   Fsh = M + mpl s¨ (1)

374

J. Geng and V. Arakelian

where M = 2(m1 + m2 ) is the total mass of the moving links of the manipulator, mpl is the mass of the payload and s¨ is the acceleration of the total mass centre. As mentioned in previous work, the adding of supplementary masses as counterweights is not desirable because it leads to the increase of the total mass, of the overall size of the manipulator, the efforts in joints, the shaking moment and the input torques. Therefore, in the suggested approach, it is proposed to minimize the shaking force via reduction of the total mass centre acceleration:  (2) maxs¨ → min s(t)

i.e. to apply an optimal control of the total mass centre of moving links that allows one to reduce the maximal value of its acceleration. For this purpose, let us consider the generation of motions in the 5R parallel manipulator through of its common centre of mass. To ensure it, let us say that the common centre of mass moves along a straight line between its initial and final positions. Consequently, the motion profile used on this path will define the values of shaking forces. Therefore, to minimize the maximum value of the acceleration of the total mass centre and, as a result, shaking forces, the “bang-bang” profile should be used [16]. Thus, by reducing the acceleration of the centre of mass of the 5R parallel manipulator, a decrease in its shaking forces will be achieved. To accomplish the shaking force balancing through the above described approach, it is necessary to consider the relationship between the end-effector and the centre of mass positions of the planar 5R symmetrical parallel manipulator with the payload and without it. 2.2 Relationship Between the End-Effector and the Centre of Mass Positions of the Planar 5R Symmetrical Parallel Manipulator with a Payload In order to control the manipulator according to the method described in the previous section, it is necessary to establish the relationship S = S(θ1 , θ2 ). Then, via obtained input angles θ1 , θ2 by means of forward kinematics determine the position of the output axis P(x, y). In [12], these relationships for the 5R symmetrical parallel manipulator have been established. Therefore, this study deals with the relationships for the manipulator with payload. For this purpose, let us establish the relationship between the common centre of mass of the manipulator with payload and its input parameters. Let us start this issue with the initial and final positions P(x, y) of the end-effector that are known: xi , yi and xf , yf . Thus, by inverse kinematics, the input angles corresponding to these positions will be determined: θ1i , θ1f and θ2i , θ2f . Let us now replace mass m1 of links 1 and 2 by two point masses located at joint centres A1 , B1 and A2 , B2 : mA1 = mA2 = m1 (l1 − r1 )/l1 and mB1 = mB2 = m1 r1 /l1 . The same for links 3 and 4: mB3 = mB4 = m2 (l2 − r2 )/l2 and mP3 = mP4 = m2 r2 /l2 . Thus, the point masses located at joints B1 , B2 can be represented by the expressions: mB = mB1 + mB3 = mB2 + mB4 and at joint P: mP = mP3 + 0.5mpl = mP4 + 0.5mpl . Then, the obtained point mass mB and mP will be presented together. Thus, two identical new masses m∗2 located on links 3 and 4 will be the sum: m∗2 = mB + mP = mB1 + m2 + 0.5mpl

(3)

Balancing of Planar 5R Symmetrical Parallel Manipulators

375

with following disposition of the centre of mass: r ∗ = l2 mP /m∗2

(4)

where, r ∗ = lB1 S3∗ = lB2 S4∗ . These masses can be substituted by one mass m∗ = 2m∗2 = mpl + 2(m2 + mB1 ), which is located in the middle of a straight line connecting two masses m∗2 . Let us denote the coordinates of the centre of mass m∗ by x∗ and y∗ (Fig. 2). Taking into account that the input angles θ1i , θ1f and θ2i , θ2f corresponding to the initial and final positions xi , yi and xf , yf of the end-effector are known, the coordinates x∗ and y∗ can be expressed as:

Fig. 2. The planar 5R symmetrical parallel manipulator with substituted point masses and its common centre of mass.

xi∗ = [mB l1 (cos θ1i + cos θ2i ) + 2mP xi ]/m∗

(5)

yi∗ = [mB l1 (sin θ1i + sin θ2i ) + 2mP yi ]/m∗

(6)

xf∗ = [mB l1 (cos θ1f + cos θ2f ) + 2mP xf ]/m∗

(7)

yf∗ = [mB l1 (sin θ1f + sin θ2f ) + 2mP yf ]/m∗

(8)

With regards to the location of the common centre of mass of the manipulator, taking into account that the point masses mA1 and mA2 are motionless, it is defined as follows: xS = x∗ m∗ /m

(9)

yS = y∗ m∗ /m

(10)

where m = m∗ + mA1 + mA2 = mpl + 2(m2 + m1 ) is the total mass of the moving links. So, having the coordinates xi∗ , yi∗ and xf∗ , yf∗ , the correspondent values of the common centre of mass of the manipulator with payload can be found: xSi , ySi and xSf , ySf . Subsequently, a straight line connecting the initial and final positions of the comment

376

J. Geng and V. Arakelian

centre mass of the manipulator can be established and its trajectory planning by “bangbang” profile can be ensured. Thus, having S = S(t), i.e. x = x(t) and y = y(t), from (9) and (10), the relationship S ∗ = S ∗ (t) can be defined, i.e. x∗ = x∗ (t) and y∗ = y∗ (t). The relationships between x∗ , y∗ and the input angles θ1 , θ2 are given in [12]. In the case of the shaking force balancing of the manipulator with payload, the same relationships can be used. Here, it is necessary to clarify that the input parameters θ1 and θ2 will be different in the operating of the manipulator with payload or without it. This is due to the fact that the locations of the common centre of mass of the manipulator with and without payload are different. To better see this, let us consider a numerical illustrative example.

3 Numerical Illustrative Example with Simulation Results To create a CAD model, the parameters of the planar 5R symmetrical parallel manipulator are the followings: l1 = 0.36 m; l2 = 0.3 m; l0 = 0.24 m; r1 = 0.18 m; r2 = 0.15 m; m1 = 0.18 kg; m2 = 0.15 kg. The trajectory of the end-effector P is given by its initial position Pi with the coordinates xi = 0.14 m, yi = 0.45 m and the final position Pf with the coordinates xf = −0.06 m, yf = 0.45 m. Two types of movements will be considered: from the initial position Pi to the final position Pf without payload and the opposite side with a payload of 4 kg. Thus, different input parameters will be obtained. The corresponding input angles are determined via inverse kinematics: θ1i = 1.2936, θ1f = 1.8557 and θ2i = 1.0813, θ2f = 1.6094. Then, the coordinates of the mass m∗ (without payload): Si∗ (0.1357, 0.3688), Sf∗ (−0.0583, 0.3830) and m∗ (with payload) are determined: Si∗ (0.1395, 0.4413), Sf∗ (−0.0598, 0.4428). Next, the coordinates of the common centre of mass of the manipulator with payload Si (0.1341, 0.4242), Sf (−0.0575, 0.4257) and without it Si (0.0987, 0.2682), Sf (−0.0424, 0.2785) are determined. Let us now connect the initial and final positions of the common centre of mass of the manipulator by the straight line and generate its trajectory planning by “bang-bang” profile. The variations of the input angles for the manipulator with payload and without it are shown in Fig. 3. Figures 4, 5 and Figs. 6, 7 show the variations of shaking forces and the shaking moments of the manipulator with payload and without it for two studied cases: 1) the displacement of the end-effector by the straight line with fifth order polynomial profile and 2) the generation of the motion via the displacement of the manipulator common centre mass by “bang-bang” profile. The simulation results show that the generation of the motion via the displacement of the manipulator centre mass by “bang-bang” profile allows the reduction of the shaking force up to 31.4% (without payload) and up to 30.7% (with payload). Moreover, although the aim of such a balancing is not the shaking moment, it also decreases up to 28.0% (without payload) and up to 30.8% (with payload).

Balancing of Planar 5R Symmetrical Parallel Manipulators

377

Fig. 3. Variations of the input angles ensuring the displacement of the common centre of mass of the manipulator without payload by the straight line with “bang-bang” profile. Curves 1 and 2 represent respectively the input angle θ1 and θ2 without payload and curves 1 and 2 represent respectively the input angle θ1 and θ2 with payload.

Fig. 4. Variations of shaking forces of the manipulator without payload for two studied cases.

Fig. 5. Variations of shaking forces of the manipulator with payload for two studied cases.

Fig. 6. Variations of shaking moments of the Fig. 7. Variations of shaking moments of the manipulator without payload for two studied manipulator with payload for two studied cases. cases.

378

J. Geng and V. Arakelian

4 Conclusions In this paper, the shaking force balancing based on the optimal motion planning has been generalized in order to apply to symmetrical 5R planar parallel manipulators with varying payload. The aim of the method consists in the fact that the manipulator is controlled not by applying end-effector trajectories but by the generation of the motion via the displacement of the manipulator common centre mass by “bang-bang” profile. For this purpose, the relationship between the end-effector and the common centre of mass positions of manipulator with varying payload is established. Then, the trajectory planning of the common centre of mass is defined as straight line having “bang-bang” acceleration motion profile. Such a motion generation allows the reduction of the maximum value of the centre of mass acceleration and, consequently, the reduction in the shaking force. It should be mentioned that such a solution is also very favorable for reduction of the shaking moment because it is carried out without adding counterweights. Although such balancing does not lead to a complete cancellation of the shaking force, but it allows one to significantly reduce it without changing the basic design of the robot. The efficiency of the suggested balancing is illustrated by numerical simulations. It should be noted that the known solutions for balancing of manipulateurs taking into account the varying payload are quite complex [17]. Significant modifications in the structure of the manipulator are necessary in order to achieve this goal. In the present paper, it has been shown that it is easy to take into account the varying payload with suggested balancing technique. It is enough to change the input parameters. Therefore, it can be attractive for industrial robot applications.

References 1. Arakelian, V., Briot, S.: Balancing of Linkages and Robot Manipulators. Advanced Methods with Illustrative Examples. Springer, Cham (2015) 2. Agrawal, S., Fattah, A.: Reactionless space and ground robots: novel design and concept studies. Mech. Mach. Theor. 39, 25–40 (2004) 3. Bayer, A., Merk, G.: Industrial robot with a weight balancing system. EP, 2301727 (2011) 4. Wang, J., Gosselin, C.: Static balancing of spatial three-degree-of-freedom parallel mechanisms. Mech. Mach. Theor. 34(3), 437–452 (1999) 5. Gosselin, C.: Gravity compensation, static balancing and dynamic balancing of parallel mechanisms. In: Smart Devices and Machines for Advanced Manufacturing, pp. 27–48. Springer, London (2008) 6. Fattah, A., Agrawal, S.: Design arm simulation of a class of spatial reactionless manipulators. Robotica 3, 75–81 (2005) 7. Van Der Wijk, V., Herder, J.: Dynamic balancing of Clavel’s delta robot. In: Computational Kinematics, pp. 315–322, Springer (2009) 8. Briot, S., Arakelian, V., Le Baron, J.: Shaking force minimization of high-speed robots via centre of mass acceleration control. Mech. Mach. Theor. 57, 1–12 (2012) 9. Arakelian, V.: Design of partially balanced 5R planar manipulators with reduced centre of mass acceleration. In: Robot Design, Dynamics and Control, pp. 113–122. Springer (2016) 10. Arakelian, V., Geng, J., Le Baron, J.-P.: Synthesis of balanced 3-RRR planar parallel manipulators. In: Proceedings of the 19th International Conference on Robotics and Computer Integrated Manufacturing (ICRCIM 2017), vol. 4, no. 9, pp. 37–43, Prague, Czech Republic (2017)

Balancing of Planar 5R Symmetrical Parallel Manipulators

379

11. Arakelian, V., Geng, J., Fomin, A.: Inertia forces minimization in planar parallel manipulators via optimal control. J. Mach. Manuf. Reliab. 47(7), 303–309 (2018) 12. Geng, J., Arakelian, V.: Design of partially balanced planar 5R symmetrical parallel manipulators via an optimal motion planning. In: Advances in Mechanism and Machine Science, pp. 2211–2220. Springer (2019) 13. Geng, J., Arakelian, V.: Partial shaking force balancing of 3-RRR parallel manipulators by optimal acceleration control of the total centre of mass. In: IFToMM WC 2019, COMPUTMETHODS, vol. 53, pp. 375–382. Springer (2020) 14. Arakelian, V., Geng, J.: Design of high-speed manipulators via optimal control of the acceleration of the total mass centre. In: Advanced Technologies in Robotics and Intelligent System, vol. 80, pp. 299–307 (2020) 15. Acevedo, M., Orvañanos-Guerrero, M.T., Velázquez, R., Arakelian, V.: An alternative method for shaking force balancing of the 3RRR PPM through acceleration control of the centre of mass. J. Appl. Sci. 10, 1–19 (2020) 16. Khalil, W., Dombre, E.: Modeling, Identification and Control of Robots, Hermes Sciences Europe (2002) 17. de Jong, J., Herder, J.: A comparison between five principle strategies for adapting shaking force balance during varying payload. In: Proceedings of the 14th World Congress in Mechanism and Machine Science, Taipei, Taiwan, 25–30 October (2015)

A Wheeled Vehicle Driven by a Savonius–Magnus Wind Turbine Marat Dosaev1(B) , Margarita Ishkhanyan2 , Liubov Klimina1 , Anna Masterova1 , and Yury Selyutskiy1 1 Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia

[email protected] 2 Federal State Institution of Higher Education, Russian University of Transport, Moscow,

Russia

Abstract. The mathematical model of a wind powered vehicle is constructed. The vehicle is located in a steady wind flow and consists of a wheeled platform carrying a horizontal axis Savonius–Magnus wind turbine. It is supposed that wheels of the platform move without slipping. Rotation of the turbine is caused by the Magnus force, which is produced by autorotating conical Savonius rotors (blades of the turbine). Rotation of the turbine is transmitted to rotation of leading wheels of the vehicle. The model of each conical Savonius rotor is represented by a pair of cylindrical rotors (a small one located closer to the turbine shaft and a large one located at greater distance from the turbine shaft). Quasi-steady model of aerodynamic load is used to describe interaction between Savonius rotors and the airflow. The steady speed of the body of the vehicle is estimated depending on parameters of the model. Two cases are discussed: up-wind and down-wind motion. Keywords: Wind powered vehicle · Magnus force · Savonius rotors · Dynamical system · Steady motion · Stability · Up-wind motion · Down-wind motion

1 Introduction Magnus type horizontal axis wind turbines are intensively studied in recent years [1–6]. Most papers in this field deals with turbines with cylindrical blades rotated by motors (e.g. [3, 4]). However Savonius rotors can be used as blades for Magnus turbines as well ([7]). This type of devices (Savonius–Magnus turbines) possesses a certain advantage: no motors are needed to support rotation of blades. Their efficiency for electricity production is discussed in [1, 5]. Advantages of using of such turbines for propulsion of wind powered vehicles were shown in [6] for the case of cylindrical Savonius rotors. Magnus type wind vehicles proved to be effective at rather low wind speeds (2–4 mps) comparable with other types of wind powered vehicles (e.g. described in [8–12]) and more effective than Savonius type wind powered vehicles (e.g. modeled in [13]) in wide range of wind speeds. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 380–386, 2021. https://doi.org/10.1007/978-3-030-58380-4_46

A Wheeled Vehicle Driven by a Savonius–Magnus Wind Turbine

381

Here we introduce the scheme of the wind powered vehicle with conical Savonius rotors used as its blades. Such vehicle can be used as a platform of an environmentfriendly mobile robot. Conical Savonius rotors were initially proposed for Magnus wind turbines in [7]. They possess certain advantages with respect to cylindrical ones, but are not well studied yet. In this paper, we construct the model where each conical rotor is substituted by two cylindrical rotors of different size. The smaller rotor is located closer to the central shaft than the larger one. This approach allows using experimental aerodynamic characteristics of cylindrical Savonius rotors that are available in the literature ([14]) and to take into account the relative flow speed distribution along the radius of each blade. The quasisteady model of an aerodynamic load is used. Two centers of pressure are introduced for each blade (they coincide with geometrical centers of two Savonius rotors). This approach of segmentation of the blade with introduction of the center of pressure for each part is rather similar to the hypothesis of flat cross-sections. Such hypothesis was used for modeling of dynamics of propeller type wind turbine in [15]. The dynamical equations of the system are analyzed. Asymptotically stable fixed points of the dynamical system correspond to regimes of steady motion of the vehicle: up-wind or down-wind motion. The speed of the body of the vehicle at such regimes is estimated depending on parameters of the model. In particular, values of the gear ratio (responsible for the transmission of rotation from the shaft of the turbine to the shaft of the leading wheels) are found which provide maximum up-wind and down-wind speed of the vehicle.

2 Description of the Model Mechanical system consists of a wheeled platform and a wind turbine located on the platform. The wheels move without slipping. The vehicle moves along the fixed horizontal axis AZ in a steady wind flow of the velocity V, which is directed against AZ axis. The axis Oz of the wind turbine rotation is parallel to the velocity of the wind. The angular speed of the central shaft is denoted by z (Fig. 1). The wind turbine has n blades. Assume that all blades are identical and that characteristics of motion are always similar for all blades. Thus, it is enough to describe parameters and characteristics of motion of one blade. The blade of the turbine is mounted at the central shaft with a cylindrical hinge. The axis Ox of this hinge is orthogonal to the central shaft. The angular speed of self-rotation of the blade in this hinge is denoted by Ωx . The blade consists of two Savonius rotors (S-rotors) that form one rigid body. Each S-rotor has cylindrical shape. The diameter of a rotor equals the height of the rotor. Further, parameters and variables related to the Srotor located at larger (smaller) distance from the central shaft have the index “1” (“2”), respectively. The following notations are used: bi is the radius of the S-rotor (hereinafter, i = 1, 2); ri is the distance from the axis Oz to the geometrical center Ci of the S-rotor. The center of mass of the blade is located at the Ox axis. Ox axis is one of the principal axes of inertia of the blade; Jx is the moment of inertia of the blade about this axis. Two other principal moments of inertia of the blade are equal to each other. Thus the moment of inertia Jz of all rotating parts of the turbine about the axis Oz is constant.

382

M. Dosaev et al.

Fig. 1. The scheme of the Savonius–Magnus wind powered vehicle.

The mechanical system has two degrees of freedom. Let M be the total mass of the system; Jw be the moment of inertia of each pair of wheels around its axis of rotation; and R be the radius of each wheel. As there is no slipping, the speed w of the platform is proportional to the angular speed of the turbine: w = kr1 z (i.e., the reduction coefficient of the transmission from the shaft of the turbine to the shaft of the leading wheels equals R−1 r1 k). Then the kinetic energy of the system is expressed by the following formula: EK =

1 2 2 2 1 1 Mr1 k z + Jw k 2 r12 R−2 2z + nJx 2x + Jz 2z . 2 2 2

Assume that the aerodynamic load upon the body of the vehicle is negligible with respect to the aerodynamic load upon the turbine. Apply the quasi-steady model ([16]) to describe aerodynamic load upon each S-rotor. Assume that the center of pressure of the S-rotor coincides with its geometrical center Ci . The aerodynamics action is described by the drag force Di , the side force Li (that includes Magnus force) and the torque T i around the Ox axis. The force Di is directed along the relative air velocity Ui of the point Ci . The force Li is orthogonal to the vector Ui . The vector Ui is represented by the difference of absolute velocity of the point Ci and the wind velocity V. The following expressions for components of the aerodynamic load are used ([14]): Ti = 0.5ρSi Ui2 bi CT (λi ), Li = 0.5ρSi Ui2 CL (λi ), Di = 0.5ρSi Ui2 CD (λi ), λi = bi x /Ui , Ui2 = V 2 + ri2 2z .

A Wheeled Vehicle Driven by a Savonius–Magnus Wind Turbine

383

Here ρ is the density of the air; Si = 4b2i is the characteristic area of S-rotor; λi is the “tip speed ratio” of the S-rotor. Functions CT (λi ), CD (λi ), CL (λi ) are dimensionless aerodynamic coefficients of the torque, the drag force and the lift force taken from experimental data. Here we use approximations (Fig. 2) of the experimental data [14]:

Fig. 2. Aerodynamic coefficients of the S-rotor.

CT (λ) = −0.118λ5 + 0.48λ4 − 0.5λ3 − 0.2λ2 + 0.25λ + 0.26; CD (λ) = 1.2; CL (λ) = 1.71λ − 0.35. Aerodynamic forces Di and Li from all 2n S-rotors produce the torque around the axis Oz:     V V r1 z r2 z Tz = nr1 L1 − D1 + nr2 L2 − D2 . U1 U1 U2 U2 Moreover, these forces produce the force acting along the Oz axis:     V V r1 z r2 z Fz = n D1 + L1 + n D2 + L2 . U1 U1 U2 U2 For further analysis, the dimensionless time and dimensionless variables are introduced: τ = 0.5J −1 nρS1 r12 Vt, ωx = V −1 b1 x , ωz = V −1 r1 z ,

384

M. Dosaev et al.

3 Equations of Motion Equations of motion of the system in dimensionless notations have the following form:   ⎧ −3 3 2 −2 2 2 2 2 ⎪ ω ˙ , C s C = a s + ω b + r r ω + b (λ ) (λ ) x T 1 T 2 ⎪ z z 2 2 1 1 ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ω˙ z = s2 + ωz2 (sCL (λ1 ) − ωz CD (λ1 )) ⎪ ⎪ ⎪

⎨  2 2 + r −2 r 2 ω2 sC (λ ) − r −1 r ω C (λ ) r b s + r1−1 b−2 2 L 2 2 z D 2 2 1 1 2 z 1 ⎪ ⎪

⎪ ⎪ ⎪ ⎪ − k s2 + ωz2 (ωz CL (λ1 ) + sCD (λ1 )) ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ −2 2 2 −1 2 ⎩ 2 − kb−2 1 b2 s + r1 r2 ωz r1 r2 ωz CL (λ2 ) + sCD (λ2 ) , s = 1 + kωz , λ1 =  a=

Jb21 Jx r12 n

ωx s2

+ ωz2

, λ2 =



b2 ωx

b1 s2 + r1−2 r22 ωz2

,

> 0, J = Mr12 k 2 + 2Jw r12 k 2 R−2 + Jz .

(1)

Further, we discuss the following

case: r1 /r2 = 2, b1 /b2 = 2, a = 10. An attracting fixed point ωx∗ ; ωz∗ of the system (1) corresponds to a steady motion of the vehicle (when ωx∗ = 0, ωz∗ = 0). The problem is to find such points depending on the parameter k.

4 Steady Motions Fixed points of the system (1) satisfy the following relations:  ⎧ 

2 2 ∗

2 2 ∗ ⎪ CT λ1 + b−3 CT λ2 = 0, 1 + kωz∗ + ωz∗ b32 1 + kωz∗ + r1−2 r22 ωz∗ ⎪ 1 ⎪ ⎪ ⎪ ⎨  

2 2 1 + kωz∗ + ωz∗ CL λ∗1 − ωz∗ + k + k 2 ωz∗ CD λ∗1 + ⎪ ⎪

⎪   ⎪ ⎪ ⎩ r −1 b−2 r b2 1 + kω∗ 2 + r −2 r 2 ω∗ 2 C λ∗ − r −1 r ω∗ + kr −1 r + k 2 r −1 r ω∗ C λ∗ = 0. L 2 D 2 z 1 1 2 2 1 2 z 1 2 z 2 1 2 1 z

Here λ∗i are steady values of tip speed ratios of S-rotors. The normalized speed v∗ of the vehicle (the speed divided by the wind speed) at steady motion is given by the formula: v∗ (k) = kωz∗ The normalized steady speed v∗ (k) for varied values of k is represented in the Fig. 3. Numerical simulation shows that corresponding fixed points of the dynamical system (1) are attracting. Negative values of k correspond to the situation when the rotation of the turbine in the “positive” direction is transmitted into the down-wind motion of the leading wheels. Maximum up-wind speed (vmax in Fig. 3) at the steady motion is about 60% of the wind speed and is achieved for reduction coefficient k ≈ 0.5. Absolute value of the maximum down-wind speed (vmin in Fig. 3) at the steady motion is about 60% higher than the wind speed and is achieved for reduction coefficient k ≈ −1.5.

A Wheeled Vehicle Driven by a Savonius–Magnus Wind Turbine

385

Fig. 3. Normalized speed of the vehicle at steady motion depending on the reduction coefficient.

5 Discussion Obtained levels of up-wind/down-wind normalized speed (Fig. 3) are lower than that for wind cars with propeller horizontal wind turbines ([8–11]). For Savonius-Magnus type vehicle, the maximum normalized up-wind speed is almost the same as that for the slider-crank type wind car proposed in ([12]). However, vehicles driven by Savonius– Magnus wind turbine are able to operate at lower wind speeds than those driven by propeller or by slider-crank wind turbine, because the autorotation of a Savonius turbine can start at wind speed from about 2 mps. It should be noted that Savonius-Magnus wind turbine allows the vehicle to reach considerably larger speed than conventional Savonius turbine (see [13]).

6 Conclusions The mathematical model of a wind powered vehicle driven by a Savonius–Magnus type wind turbine is introduced. The novelty of the model comparing with [6] is that it describes a case of conical blades of the turbine. In this model, conical Savonius rotor that represents a blade of the turbine is substituted by a pair of cylindrical Savonius rotors of different size. This approach allows taking into account the relative flow speed distribution along the radius of each blade. The model is represented by a second order dynamical system. Fixed point of this system corresponds to steady motion of the vehicle. These fixed points are studied depending on parameters of the model. In particular, the maximum speed of the vehicle for up-wind and down-wind motion is estimated. Values of reduction coefficient that provide maximum levels of speed are obtained. At the next step of the research, the laboratory prototype of the wind car is to be constructed; the model is to be verified basing on experimental tests with such a prototype in a wind tunnel.

386

M. Dosaev et al.

This work was partially supported by the Russian Foundation for Basic Research, projects NN 18-01-00538, 19-31-90073.

References 1. Akira, I., Kawashima, S., Nishizawa, Y., Ushiyama, I., Komatinovic, N. A study on Savonius type Magnus wind turbine. In: Europe Premier Wind Energy Event (2007) 2. Sun, X., Zhuang, Y., Cao, Y., Huang, D., Wu, G.: A three-dimensional numerical study of the Magnus wind turbine with different blade shapes. J. Renew. Sustain. Energy 4(6), 063139 (2012) ˇ c, M., Deur, J.: Operating cycle optimization for a Magnus effect-based 3. Milutinovi´c, M., Cori´ airborne wind energy system. Energy Convers. Manage. 90, 154–165 (2015) 4. Marzuki, O.F., Mohd Rafie, A.S., Romli, F.I., Ahmad, K.A.: Magnus wind turbine: the effect of sandpaper surface roughness on cylinder blades. Acta Mech. 229(1), 71–85 (2017). https:// doi.org/10.1007/s00707-017-1957-6 5. Ishkhanyan, M.V., Klimina, L.A., Privalova, O.G.: Autorotation motions of a turbine coursed by the magnus effect. AIP Conf. Proc. 1959, 030010 (2018) 6. Dosaev, M., Ishkhanyan, M., Klimina, L., Privalova, O., Selyutskiy, Y.: Wind car driven by the magnus force. In: ROMANSY 22 – Robot Design, Dynamics and Control. Proceedings of the 22nd CISM IFToMM Symposium, June 25–28, 2018, vol. 584, pp. 189–195. Springer, Cham (2019) 7. Savonius, S.J.: Rotor adapted to be driven by wind or flowing water. U.S. Patent No. 1697574 A (1929) 8. Bauer, A.B.: Faster than the Wind. In: First AIAA Symposium on Sailing (1969) 9. Korepanov, V.: Four times faster than the wind. Catal. J. AYRS 18, 21–27 (2004) 10. Wilson, J.C.: Speed downwind vs. gear ratio in a Bauer vehicle. Catal. J. AYRS 28, 18–19 (2007) 11. Dosaev, M., Klimina, L., Selyutskiy, Y.: A vehicle driven upwind by the horizontal axis wind turbine. In: EuCoMeS 2018. Proceedings of the 7th European Conference on Mechanism Science. Mechanisms and Machine Science, vol. 59, pp. 155–161. Springer, Netherlands (2019) 12. Klimina, L., Dosaev, M., Selyutskiy, Yu.: Asymptotic analysis of the mathematical model of a wind-powered vehicle. Appl. Math. Model. 46, 691–697 (2017) 13. Selyutskiy, Y.D., Klimina, L.A., Masterova, A.A., Hwang, S.S., Lin, C.H.: Savonius rotor as a part of complex systems. J. Sound Vib. 442, 1–10 (2019) 14. Bach, G.: Untersuchungen über Savonius-Rotoren und verwandte Strömungsmaschinen. Forschung auf dem Gebiet des Ingenieurwesens A 2(6), 218–231 (1931) 15. Dosaev, M.Z., Klimina, L.A., Lokshin, B.Ya., Selyutskiy, Y.D.: On wind turbine blade design optimization. J. Comput. Syst. Sci. Int. 53(3), 402–409 (2014). https://doi.org/10.1134/S10 6423071403006X 16. Dosaev, M.Z., Samsonov, V.A., Selyutskii, Y.D., Lu, W.-L., Lin, C.-H.: Bifurcation of operation modes of small wind power stations and optimization of their characteristics. Mech. Solids 44(2), 214–221 (2009)

Development of an Off-board Vision-Based Control for a Micro Aerial Vehicle Jorge Solis1,2,3(B)

, Christoffer Karlsson1 , and Kristoffer Richardsson4

1 Department for Engineering and Physics, Karlstad University, Karlstad, Sweden

[email protected] 2 Research Institute for Science and Engineering, Waseda University, Tokyo, Japan 3 Department for Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan 4 Bitcraze AB, Malmö, Sweden

Abstract. Our research aims to develop an intelligent robot vehicle with multimodal locomotion capabilities for the forest monitoring purpose. Due to the complexity of the proposed research, in this paper, we proposed a visual-based control system; capable to detect fiducial markers and pose estimation even with limited capabilities and transmission losses, in order to hover, navigate and fly to the desired target location while remaining stable without adversely affecting the effective flight time due to additional sensors and computation. Based on the experimental results, the MAV was able to detect the fiducial markers with a success ratio of about 92.8% as well as to land on the desired location within a radius of about 10 cm from the center-point of the landing pad, with a reduction of the effective flight time of about 28%. Keywords: Micro aerial vehicles · Visual-based control · Kalman filter

1 Introduction 1.1 Background Forestry is still a demanding area for robot technology, where reliable perception and measurement of essential objects are required to develop more enhanced autonomous or tele-operated operations. So far, mobile robots have been used, for specific environmental conditions, as fundamental data gathering tools with a high-resolution sampling in both space and time ([1], etc.). More recently, the introduction of unmanned aerial vehicles (UAVs) has become economic and much more widely used for environmental monitoring for large areas [2]. The development of hybrid locomotion autonomous robots, efforts drawing upon inspiration from biology, has had resulted in remarkable progress towards autonomous robotic platforms. However, there are still several technological issues, which have impeded the development of truly autonomous robotic platforms capable of motion in more than one environmental medium. In particular, one of the main challenges of transitioning between combinations of aerial [3], aquatic [4], and terrestrial [5] modes of locomotion present an array of issues ([6], etc.). © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 387–395, 2021. https://doi.org/10.1007/978-3-030-58380-4_47

388

J. Solis et al.

Therefore, the authors have proposed to incorporate and develop the concept of biologically inspired design approaches to enable the application to an intelligent Robot Vehicle with multimodal locomotion capabilities (iRoVm) for the forest monitoring purpose. Due to the complexity of the task; as a first approach, in this paper we focused to design and implement an off-board control system based on visual input for a micro aerial vehicle (MAV). By integrating a camera and wireless video transmitter onto the MAV platform, we are able to achieve autonomous navigation and landing in relatively close proximity to the dedicated target location and the effective flight should not be adversely affected by more than 50% due to additional sensors and peripherals. 1.2 Related Work Falanga et al. [7] developed a quadrotor system capable of achieving autonomous navigation and landing on a moving platform using only on-board sensing and computing. Lange et al. [8] presented a similar concept using optical flow and vision-based detection, where the marker depicting the target landing location consists of several concentric white rings on a black background. Foster et al. [9] has demonstrated a method for achieving autonomous hovering using a monocular SLAM-system for the Crazyflie. With this method, the Crazyflie is able to remain stable whilst in flight without the need of using any external sensors. However, the system relies on pose estimation through the Kinect RGB-D sensor and therefore, it must be configured as an external set-up, meaning that the control system requires data from sensors external to the MAV platform in order to achieve autonomous waypoint flying. However, most of the proposed approaches depends still on human inputs, advanced calibration procedures and do not take into account the effect on the effective flight time due to the use of additional sensors and peripherals.

2 Vision-Based Control for a Micro Aerial Vehicle 2.1 Hardware The commercially available Crazyflie 2.0 from Bitcraze AB was used [10]. The Crazyflie 2.0 offers the possibility to connect expansion boards to the headers. The Flow deck V2 is an expansion board that includes an optical flow sensor and acts as a complement to the IMU [10]. In order to be able to localize the marker and determine the relative pose of the quadcopter, it is necessary to add an additional sensor. For this purpose, the camera and transmitter module from the company Eachine was used [11] and integrated on-board of the Crazyflie. 2.2 Relative Pose Estimation and Fiducial Marker Detection As a first approach, the ArUco module developed by Muñoz and Garrido [12] was employed as the library for target detection and pose estimation. The library is composed of a set of dictionaries of different markers, each with its own unique identifier but with the same characteristic design. An ArUco marker is a synthetic square planar marker with an inner binary identifier matrix contained within a thick black border (Fig. 1a).

Development of an Off-board Vision-Based Control

389

When the four corners of the desired marker have been detected, the pose of the camera can be estimated through a homography transformation function. This is done by using the ArUco’s function estimatePoseSingleMarkers, which it uses the direct linear transformation to initialize the camera pose and then the Levenberg-Marquard optimization method to minimize the back-projection error. As an output, it returns one vector for the translation and another one for the relative rotation. Then, the next step is to determine the rotation matrix, Rzyx , which holds the rotation between the two reference frames in a similar fashion as indicated in Eq. (1). By using Rodrigues formula shown in Eq. (2), where uˆ is the skew-symmetric matrix representation of the vector u, it is possible to compute uˆ and φ by using Eq. (3) and (4). τ is the sum of the scalar values in the main diagonal of the rotation matrix R (τ = r11 + r22 + r33 ). ⎡

⎤ coscos sinsincos − cossin cossincos + sinsin ⎢ ⎥ R(, , ) = ⎣ cossin sinsinsin + coscos cossinsin − sincos ⎦ −sin sincos coscos

(1)

Rot(u, φ) = Icosφ + uuT (1 − cosφ) + uˆ sinφ

(2)

uˆ =

1  R − RT 2sinφ

cosφ =

τ −1 2

(3) (4)

The process of detecting the markers in the images taken by the camera is comprised of two main steps. The first step involves image segmentation through local adaptive thresholding and extraction of contours so that we can find square shaped candidates of markers in the image. Those rectangular shapes found in the thresholded image that do not approximate to a square are disregarded and we proceed to the second step. Once we have found a set of candidates for markers in the image, we analyse the inner binary matrix of each candidate by extracting the white and black bits to determine which of them valid markers are. This is done through computing the homography matrix and thresholding using the Otsu’s method [12] so that perspective projection is removed and we get a binarized image containing a grid of binary values. The bits are analyzed to determine if the marker belongs to the specific dictionary. However, because of the noise introduced by e.g., the limited capabilities of the camera, transmission losses, etc., we require a method for estimating the relative pose of the MAV. For this purpose, in order to improve the pose estimation by the ArUco library, we have proposed to add a Kalman filter, firstly introduced by Kalman [13], which addresses the problem of predicting the state of a dynamical system at discrete time step, k, given measurements from the current state at time step k − 1 and its uncertainty matrix. As a first approach in our project, we have proposed to use the linear Kalman filter, which it is a discrete time algorithm consisting of a prediction and a measurement update step that can be run independently from one another. If we assume a current state of our system, xk , containing elements describing its position, p, and velocity v, we can define the Eq. (5), where k indicates the current discrete time step. The linear Kalman filter will then assume a correlation between the elements contained in xk , captured by a covariance matrix, Pk .

390

J. Solis et al.

It will then produce an estimate xˆ k , from the previously estimated state xˆ k−1 , from time step k for the conditional probability that the error between the measured and estimated state are Gaussian distributed with a mean value μ and variance σ 2 . The measurement update step then feeds back the measurement into the a priori estimate to obtain an a posteriori estimate. xk = (p, v)

(5)

We make this possible by taking each point in our original estimate and move it to a new predicted point, governed by the parameters that dictate how the system would behave if that original estimate was correct. The prediction step can be represented by a state transition matrix, Fk , as it is shown in Eq. (6). By defining the estimation as Eq. (7) when considering that we expect an acceleration, α, this give us the state estimation as Eq. (7). Bk is the control matrix and uk is the control vector used as input to the system. In order to take into account the presence of external disturbance such as noise with covariance Q (assuming to be normal distributed), we add them after every prediction step by using Eq. (9).

 1 t Fk = (6) 0 1 pk = pk−1 + tvk−1 + 21 α t 2 (7) xˆ k = vk= vk−1 + α t xˆ k = Fk xˆ k−1 + Bk uk

(8)

Pk = Fk Pk−1 FkT + Qk

(9)

The final part of the algorithm is the measurement part. As we receive new data, we stored into a vector, z. These measurements may contain some level of noise, here denoted as R, so we can compute zk as Eq. (10). The matrix H is a general matrix of which we multiply the state into to convert it into a measurement matrix. By incorporating the state estimation equation as Eq. (11), where K is known as the Kalman filter, the final equation for the update state can be defined as Eq. (12) and (13), where K  is determined as Eq. (14). zk = Hxk + Rk

(10)

xˆ k+1 = Fxk + Ky

(11)

 xˆ k = xˆ k + K  zk − Hk xˆ k

(12)

Pk = Pk − K  Hk Pk

(13)

 −1 K  = Pk HkT Hk Pk HkT + Rk

(14)

Development of an Off-board Vision-Based Control

391

Notice that if the desired marker has not been detected for n number of frames in a row, the velocity for each state element is exponentially decreased such that for each time step, kn ≤ nmax , where kn denote a time step with an undetected marker, the updated estimation was computed as Eq. (15), where α is the diminishing factor. xˆ k n (vk ) = xˆ k n (vk−1 )α

(15)

2.3 Control Architecture As it is shown in Fig. 1a, the computer acting as the ground station receives the images from the RGB-camera and it can perform a two-way communication with the Crazyflie 2.0 through the Crazyradio PA. Through the ArUco library, each grabbed image frame is processed in the computer and once a marker has been detected, the pose of the quadcopter relative to that marker in that image frame is calculated and is fed to a Kalman filter. The Kalman filter makes an optimal estimation of the relative pose of the MAV. An error signal is then computed and the PID controllers output the commanded control signal in order to align the MAV pose towards the marker. As a first approach, the control signals from the off-board PID controllers comprises the relative yaw ()-angle and the translation in x and z-direction (Fig. 1b). The on-board controller consists of two PID controllers in cascade. As a first approach, the AMIGO method was chosen as the primary tuning method (the resultant gain control parameters are K p = 1.0, K i = 0.10, K d = 0.045 and K p = 5.10, K i = 5.40, Kd = 0.62 respectively).

Fig. 1. a) Representation of the physical system; b) Proposed control system.

3 Experiments and Results 3.1 Pose Estimation and Fiducial Marker Detection In order to evaluate the pose estimation based on the linear Kalman filter, a short tracking sequence was defined. For this purpose the fiducial marker was moved in various angles in front of the MAV whilst it was hovering and was given attitude control signals such that it tries to always have its y-axis pointing collinear with the z-axis of the reference

392

J. Solis et al.

frame of the marker, radiating out from its center. The experimental results are shown in Fig. 2. As can be observed, at the time t ≈ 12.5 s, the detection of the marker is lost for approximately one second and the Kalman filter predicts the movement of the marker given the velocity at this time which gets reduced by the factor α for each iteration until the maximum number of frames, nmax has been reached.

Fig. 2. a) Experiment for estimation along the yaw (); b) Experiment for estimation of marker center along the x-direction.

As for the fiducial marker detection, an experiment for evaluating the performance of the detection algorithm aimed to examine at which distance, x, the fiducial marker can be detected. The experiment was conducted by placing the marker at various distances, x ∈ [10, 220] centimeters away from the camera. In order to keep the results of the experiments consistent, the marker used is characterized by a 200 mm × 200 mm square with a 6 × 6 internal matrix and has the identifier ID = 24. The experimental results are shown in Fig. 3, where the detection success of the marker achieved the 92.8% (39 out of 42). As it can be observed, the pose estimation algorithm is able to estimate the distance between the camera and the marker with an error about 2 cm when the marker is placed at a distance of about 1 m.

Development of an Off-board Vision-Based Control

393

Fig. 3. Actual distance between the camera and the marker, versus the distance calculated by the marker detection and pose estimation function.

3.2 Target Landing In order to test the performance for target landing, the marker was placed away from the MAV with a constant angle α (Fig. 4a). Both the marker and the quadcopter were placed on the same initial height and the MAV was programmed to take off and hover above ground at a height of 30 cm. By measuring the point of where the MAV landed in relation to the desired landing location (Fig. 4b), we can evaluate the accuracy of the vision system. The experiment was conducted with five iterations, keeping the distance x and the angle α as close to constant as possible between each trial. As it is shown in Fig. 5, the MAV is capable of landing close to the center of the landing platform with a standard deviation of about σ ≈ 5.5 cm in both x- and y-direction.

Fig. 4. a) The Crazyflie 2.0 was placed at a distance of 1.5 m away from the marker with an angle α = 30o ; b) ArUco marker with ID = 24 from the 6 × 6 dictionary used as target. Below the marker lies a measurement pad on which the MAV should land.

394

J. Solis et al.

Fig. 5. Each point, Pi on the diagram correspond to the landing point for each attempt.

3.3 Battery Life Characterization The Crazyflie 2.0 has a flight time of around 5 min without any additional equipment [10]. For our purposes, the flight time was measured first without the vision system modules and secondly with the modules mounted and turned on. As it is shown in, Fig. 6. Based on experimental results, it can hover in place for roughly 6.7 min until the battery voltage level decreases below 3.0 V.

Fig. 6. Experimental results for the battery characterization.

4 Conclusions We presented an off-board vision-based control system for the Crazyflie 2.0. By means of the ArUco library, each obtained image frame is processed and once a fiducial marker was detected, the pose of the MAV is calculated and optimized by a linear Kalman filter. By means of two PID controllers in cascade, the MAV could align and orient itself towards the marker. Based on the obtained experimental results, we can conclude that the implemented vision system allows the MAV to detect successfully 92.8%, navigate towards and land in proximity to the target marker (within a radius of 10 cm).

Development of an Off-board Vision-Based Control

395

References 1. Dunbabin, M., Roberts, J., Usher, K., Corke, P.: A new robot for environmental monitoring on the great barrier reef. In: Australian Conference on Robotics and Automation (2004) 2. Hodgkinson, B.: Environmental monitoring with k-means error reduction using UAVs controlled by a fluid based scheme. In: Workshop on Robotics for Environmental Monitoring (2012) 3. Bachmann, R., et al.: A biologically-inspired micro sensor platform capable of aerial and terrestrial locomotion. Mech. Mach. Theor. 44, 512–526 (2009) 4. Ijspeert, A.J., et al.: From swimming to walking with a salamander robot driven by a spinal cord model. Science 315(5817), 1416–1420 (2007) 5. Daltorio, K.A., et al.: Mini-Whegs TM climbs steep surfaces using insect-inspired attachment mechanisms. Int. J. Robot. Res. 28(2), 285–302 (2009) 6. Low, K.H.: Mechatronics design and locomotion implementation of robotic fish with modular flexible fins. J. Syst. Control Eng. 22(1), 295–309 (2007) 7. Falanga, D., et al.: Vision-based autonomous quadrotor landing on a moving platform. In: 15th International Symposium on Safety, Security and Rescue Robotics, Conference, pp. 200–207 (2017) 8. Lange, S, Sunderhauf, N., Protzel, P.: A vision based onboard approach for landing and position control of an autonomous multirotor UAV in GPS-denied environments. In: Proceedings of the International Conference on Advanced Robotics, pp. 1–6 (2009) 9. Forster, C., Pizzoli, M., Scaramuzza, D.: SVO: fast semi-direct monocular visual odometry. In: IEEE International Conference on Robotics and Automation, pp. 15–22 (2014) 10. Bitcraze AB. https://store.bitcraze.io/products. Accessed 6 Feb 2020 11. Eachine - FPV VTX Camera. https://www.eachine.com/FPV-SYSTEMS-c-153.html. Accessed 9 Feb 2020 12. Ramirez, F.R., Salinas, R.M., Carnicer, R.M.: Speeded up detection of squared fiducial markers. Image Vis. Comput. 76, 38–47 (2018) 13. Welch, G., Bishop, G.: TR 95-041: an introduction to the Kalman Filter. University of North Carolina, Department of Computer Science, pp. 1–16 (2006)

Stiffness Optimization of Delta Robots Christian Mirz1(B) , Olaf Uzsynski2 , Jorge Angeles3 , Yukio Takeda4 , and Burkhard Corves1 1

Institute for Mechanism Theory, Machine Dynamics and Robotics, RWTH-Aachen University, Aachen, Germany [email protected] 2 Institute for Automotive Engineering, RWTH-Aachen University, Aachen, Germany 3 Robotic Mechanical Systems Laboratory, McGill University, Montreal, Canada 4 Mechanical Systems Design Laboratory, Tokyo Institute of Technology, Tokyo, Japan

Abstract. Dynamic manipulation tasks require lightweight yet stiff robotic systems, to achieve high efficiency and precision at the same time. The most widely spread parallel robot designed for these tasks is the Delta robot. To decide on the dimensional parameters for these systems, best meeting the foregoing requirements, this paper focuses on the optimum dimensional synthesis, combing kinematic, dynamic and stiffness design criteria. In addition to the kinematic parameters of the robot, the shape of the proximal links is optimized based on two different designs. One carrying a solid section and one a hollow section profile, both paradigms of the sections found in the market. A case study, based on a common application of the Delta robots, is used for a critical discussion on the performance of both proximal link designs.

Keywords: Delta robot

1

· Stiffness optimization · Shape optimization

Introduction

Driven by rising energy costs and the need for industrial automation, the demand for efficient and sustainable robotic systems is steadily growing. Dynamic manipulation tasks, as found in the packaging industry, demand lightweight, stiff robotic systems, with a high-payload-to weight ratio, to achieve both energy efficiency and high positioning accuracy. Given these requirements, parallel robots are the best choice because of their architecture with a ground based actuation system and hence, low moving mass. The most widely spread parallel robot designed for manipulation tasks is the Delta robot. Along these lines, this paper is concerned with the dimensional synthesis and optimization of Delta robots. Since its invention, several optimization strategies Supported by the German Academic Exchange Service (DAAD). c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 396–404, 2021. https://doi.org/10.1007/978-3-030-58380-4_48

Stiffness Optimization of Delta Robots

397

have been proposed. Early strategies focused on kinematic design objectives e.g. maximum actuator velocity [4]. Later these were complemented with dynamic design objectives, such as the average power requirement. For a comprehensive overview the reader is referred to [4]. In recent years new approaches combining dynamic and stiffness objectives have been reported. Zhang and Song [15] proposed a dynamic index taking into account the ratio of average power requirement and total moving mass, while constraining the first natural frequency. Courteille et al. [10] proposed a multi-objective optimization considering global stiffness indices, derived on the basis of a structural analysis, while constraining the moving mass to guarantee for sufficient dynamic performance. This paper addresses the combination of kinematics, dynamics and stiffness optimization. This allows not only the optimization of the geometric parameters but also a shape optimization of individual links. Furthermore, precise knowledge of the link geometry allows a more accurate estimation of dynamic loads and therefore, superior performance. Here, the focus is on the shape optimization of the proximal links, as it has a major influence on the overall robot compliance [9]. Contrary to previous works, the cross-section of the proximal links is not restricted to a rectangular or a cylindrical shape. Instead, an I-beam and a conical tube with an oval cross-section, as described in Sect. 2, are used. The optimization is based on an elastostatic stiffness analysis, taking into account four different stiffness indices, as in Sect. 3. Finally, a case study, based on a common application of Delta robots, is included.

2

Formulation of the Optimization Problem

The proposed optimization algorithm is based on the kinematics and dynamics optimization presented in [4] by Brinker et al. In addition to the kinematic design parameters, such as the radii of the base (rb ) and moving platform (rp ), the lengths of the proximal links (lp ) and distal links (ld ), the distance between the centers of the spherical joints (ds ), as well as the workspace offset (z0 ), the geometry parameters of the proximal links will be optimized for two geometries. To represent the most common geometries found in industry, one profile with a solid I-shaped and one with a hollow oval shaped section is chosen (Fig. 1). In both cases, the wall thickness is assumed to be constant over the length and the fixation points of the spherical joints being set at 10 mm below the lower end of the proximal links’ front faces, to avoid collision with the distal links. The robot’s stiffness properties are calculated using the Virtual Joint Method [9], using analytical models for the proximal links. The compliance matrices are derived using Timoshenko beam theory. In addition to the compliance of the proximal links (made of aluminium), the longitudinal compliance of the distal links (CFRP) and the torsional compliance of the gearbox are taken into account, both obtained by measurements on an actual robot. The moving platform is assumed to be a disk with radius rp made of aluminium with 10 mm thickness. Using this model, a compliance matrix C ∈ R6×6 , representing a linear mapping between external loads acting on the platform to their corresponding deflections, is calculated. To evaluate and optimize the stiffness of different robots, a

C. Mirz et al. a)

rb

b)

lp

r2

r1

t

R1

lp

b1

b2 t

h2

rp

c)

h1

Z2 Z 1

ld

z0

h1

ds

lp h2

398

R2

Fig. 1. Kinematic parameters of a Delta parallel robot and its workspace (gray) (a) and geometry parameters of the hollow (b) and solid section (c) proximal link profile

scalar index is to be defined from this matrix, representing the overall stiffness performance. A detailed discussion on the stiffness indices is included in Sect. 3. For the kinematics optimization, the input (ITI) and output transmission index (OTI), as well as their product, the total transmission index (TTI), are evaluated to assess the quality of motion-and-force transmission. For further information about the kinematic performance criteria it is referred to Brinker’s work [3]. These criteria are calculated at 62 points (as per Fig. 2b). To lower computational costs, these criteria are evaluated in one third (120◦ ) of the prescribed workspace, by virtue of the robot symmetry. The height and radius of the upper-cylindrical section of the workspace are defined as Z1 = 0.18 m and R1 = 0.6 m. The conical section has a height of Z2 = 0.12 m and a radius of R2 = 0.45 m. These dimensions are defined on the basis of a market analysis [3].

Fig. 2. Paradigm trajectories used to evaluate the robot dynamics (a). Workspace with 62 evaluation points for the assessment of the kinematic performance (b)

Furthermore, three dynamic design objectives are taken into account; the maximum power requirement (Pmax ) of a single motor, the maximum combined

Stiffness Optimization of Delta Robots

399

 power requirement of all three motors ( Pmax ), acting as an indicator for the energy-efficiency and the maximum motor torque (Tmax ). The latter is of great importance for the motor selection, since it is correlated to the size and cost of motors and gears. All dynamic design objective are evaluated for eight paradigm trajectories, as per Fig. 2a, following the Adept cycle. Four trajectories with 300 mm and four with 700 mm translational motion are defined, all with 25 mm upstroke. The corner smoothing is conducted using Lam´e curves as suggested by Barnard et al. [2]. The correlation between the path and displacement over time is established using a 4-5-6-7 interpolating polynomial in order to guarantee smooth first-, second-, and third-order derivatives, as proposed by Gauthier et al. [11]. The cycle time is set to 0.1 s and 0.2 s, respectively, both with a payload of 1 kg. In addition to the foregoing design objectives, the workspace ratio, i.e. the quotient of the work- and installation space, is taken into account. The method used for the dynamic modelling is as proposed by Brinker et al. [5]. Kinematics, dynamics and stiffness design objectives behave antagonistic. To obtain all possible trade-off solutions, the design task is formulated as a multiobjective optimization problem, with multiple nondominated solutions, known as Pareto optimal candidates, being calculated. In order to avoid local minima and potentially find a global optimum, a stochastic optimization approach is used, i.e. R Prior optimization, an algorithm based on NSGA-II provided by MATLAB. a constraint satisfaction problem is solved, to determine feasible bounds for the kinematic parameters (rb , rp , lp , ld , ds , z0 ) and to improve the convergence behavior. For the details, the reader is referred to the pertinent literature [3].

3

Stiffness Evaluation Criteria

In general, industrial robots show coupling between translational and rotational deflections caused by forces and torques, respectively. Thus, the compliance matrix C is densely populated and dimensionally inhomogeneous in its four blocks representing deflections δx due to forces f , in C11 and torques t, in C12 , as well as the corresponding rotations δϕ given by C21 and C22 , respectively.     C11 [m/N] C12 [1/N] δx [m] w, (1) δs = = Cw = T δϕ [-] [1/N] C22 [1/Nm] C12 T  Here, w = f T , tT denotes the vector of external loads acting on the moving platform. For dimensionally homogeneous matrices, the most common stiffness indices are the eigenvalues and eigenvectors, the singular values, the determinant and matrix norms [6,8,10,12]. For a comprehensive overview the reader is referred to [7,17]. However, in case of dimensionally inhomogeneous matrices all of the above mentioned indices bear no physical meaning [17]. To address this problem, several approaches have been introduced, some of which define a dimensionally inhomogeneous index, others transform or normalize the stiffness matrix. However, this usually requires the definition of a characteristic length, which has been difficult to agree upon [7].

400

C. Mirz et al.

An intuitive and physically meaningful way is to investigate the stiffness/ compliance (dimensionally homogeneous) block matrices separately [14]. In this case all indices defined for homogeneous matrices are applicable. Here, the singular values of the individual compliance block matrices indicate the magnitudes of the principal axes of the compliance ellipsoid, i.e. the magnitude of a deflection due to a unit force/torque, and the corresponding left-singular vector indicates the direction of the principal axis/deflection. Additionally, the stiffness properties can be assessed by calculating the displacements resulting from unit loads. However, using this methodology typically leads to an overestimation of the stiffness properties, since in general, the direction of the lowest stiffness is unknown. A mathematical way to address the dimensional inconsistency of the stiffness matrix was proposed by [1,13,16,18]. Here, a generalized eigenvalue problem based on screw theory is introduced [1] Kvk,i = λk,i Γ vk,i

with

i = 1, ..., 6.

(2)

The vector vk,i is a six-dimensional generalized eigenvector in screw coordinates, representing the displacement screw and K denotes a R6×6 stiffness matrix. For a dimensionally-correct formulation the permutation matrix Γ is introduced, [1]   0 I3×3 . (3) Γ = I3×3 0 In the context of screw theory, a normalization of the eigenvector is meaningless, so is the corresponding eigenvalue. Instead, the product of both is scaled in a way to obtain a unit screw [1]. KA v$,k,i = λ$,k,i Γ v$,k,i i = 1, ..., 6     si δϕk,i . λ$,k,i v$,k,i = λ$,k,i 0 = λk,i vk,i = λk,i si δxk,i

(4) (5)

The unit eigenscrew is denoted by v$,k,i and λ$,k,i is the corresponding eigenvalue. For further information the reader is referred to [1]. The generalized eigenscrew calculation associated with the compliance matrix is calculated likewise: Cv$,c,i = λ$,c,i Γ v$,c,i

with

i = 1, ..., 6.

(6)

The normalised eigenvalues λ$,K,i are referred to as eigenforces [1]. The generalized eigenvalues λ$,c,i of the compliance matrix (CIE), with units of [1/N], are the equivalents to the eigenforces of the stiffness matrix. As the eigenvalues of homogeneous matrices, the screw analogues can be used as an stiffness index. They are evaluated at the same points as the aforementioned kinematic indices. As the kinematic indices, the stiffness indices are not only depending on the dimensions of the robot, but also on the position, and are therefore local indices. As a global index, the mean values, the global maximum/minimum over the workspace, or the ratio of maximum to minimum value can be used [6,17]. The latter is a measure of the stiffness anisotropy.

Stiffness Optimization of Delta Robots

4

401

Optimization and Results

7 1 6 5

0.8 0.4

10−5

0.2 T T I [-]

Pmax [kW]

Tmax [kNm]

In total, 28 optimization runs, combining at least one kinematic, dynamic and stiffness criterion were conducted. Of these, 14 runs were based on each, the hollow and solid section profile. As kinematic index, the global minimum of either the OTI,  ITI or their product, the TTI, was used, in combination with the Tmax , Pmax , as dynamic index. As for the stiffness indices, the maximum Pmax or generalized eigenvalue λ$,c,i (CIE) of the compliance matrix or the maximal singular values of the upper compliance block matrices C11 (SV Dxf ) and C12 (SV Dxm ) are considered. In combination, the latter describe the translational compliance of the robot.

10−4 CIE [N−1 ]

Fig. 3. Pareto frontiers of optimization with kinematics, dynamics and stiffness objectives, considering solid (black circled) and hollow section profile proximal links

Figure 3 shows the resulting Pareto frontier. Apparently, the stiffness of the I-shaped section profile (circled in black) is significantly lower than the stiffness of the hollow section, ovally-shaped profile. Both Pareto frontiers show a similar shape. As expected, the limits of the kinematic indices are not influenced by the choice of the proximal link profile. However, choosing a hollow section profile allows for lower actuator torques while maintaining superior stiffness and kinematic properties, when compared to the solid section profile, as indicated in Fig. 3 by a lower gradient of the Pareto frontier towards higher T T I. Table 1. Optimized kinematic parameters of the fittest robots, with respect to different design objectives

Objective

Kinematic Parameters [mm]

TTI Pmax Tmax CIEmax

Hollow Solid Hollow Solid Hollow Solid Hollow Solid

rb

rp

lp

ld

z0

ds

h1

h2

t

275 248 244 276 291 261 234 236

76 76 154 97 53 53 81 68

399 412 288 292 342 309 263 272

1072 1101 1043 981 935 949 1056 1083

-954 -979 -1006 -878 -800 -838 -960 -984

188 185 197 133 105 97 200 152

74 122 69 103 84 19 69 122

49 19 60 22 39 12 57 14

9 9 10 10 6 6 10 10

r1 w1 33 75 37 72 23 58 40 77

r2 w2 32 68 35 58 28 35 40 69

402

C. Mirz et al.

In general, optimizing the kinematic indices leads to robots with long proximal and distal links and a comparably small platform radius (cf. Table 1). For a detailed explanation, the reader is referred to Brinker’s work [3]. This in turn, causes a decreasing stiffness of the robot, as well as a higher power requirement and motor torques due to increased leverage and moving mass. Minimizing the compliance of the robot leads to a compact design with short proximal links. Optimizations with the hollow section profile result in larger ds than those, with the solid section profile. This leads to a larger leverage of the distal links about the platform center, thereby increasing the torsional stiffness. At the same time, it results in higher torsional/bending loads on the proximal links. The hollow profile excels, due to a higher torsional stiffness, since that of a thinwalled solid profile is only approximately as high as that of the developed sheet. Gears

46%

39%

15% Torsion

Distal Links

68% 13%

Bending y-Direction

(a) Open Section Profile

Proximal Links 59%

85% 12%

3%

Bending z-Direction

24% 18%

Others

(b) Closed Section Profile

Fig. 4. Contribution of each element in the kinematic chain of the Delta robot to a deflection caused by a unit force, for both, solid and the hollow proximal link profiles

This becomes apparent when analysing the contribution of each element in the robot’s kinematic chain, to a deflection caused by a unit force acting on the moving platform. Looking at the stiffest robot with solid profile links, a major share of 39% of the deflection is caused by the compliance of the proximal links (see. Fig. 4(a)). The largest share, with 68% of the proximal links’ deflection, is caused by torsion. In the case of the stiffest hollow profile (Fig. 4(b)) only 3% of the deflection is caused by the proximal links. Again, the highest contributor to the deflection by the proximal links is torsion with 59%. This holds for all robots analyzed, thereby showing the importance of a sufficient torsional stiffness. In the presented optimization, the average robot with a hollow profile has an oval cross-section (h1 = 65 mm, h2 = 55 mm) and a slightly conical shape (r1 = 38 mm, r2 = 33 mm) with a wall thickness of t = 9.3 mm. Purely cylindrical and conical designs, as often found in industry, did not show a better performance.

Stiffness Optimization of Delta Robots

5

403

Conclusions

In this paper a combined kinematics, dynamics and stiffness optimization was proposed, using two different designs for the proximal link. One design is based on an solid I-shaped section and one on a hollow section profile, both paradigms of the sections found in the market. The analysis of the optimization results revealed that choosing a hollow section profile will lead to a superior design in both stiffness and dynamic properties compared to the solid section profile. Furthermore, the higher torsional stiffness of the hollow profile allows for increased kinematic performance while maintaining the stiffness characteristics.

References 1. Angeles, J.: On the nature of the cartesian stiffness matrix. Ingenier´ıa mech´ anica, tecnolog´ıa y desarrollo 3(5), 163–170 (2010) 2. Barnard, C.J., Briot, S., Caro, S.: Trajectory generation for high speed pick and place robots. Eng. Syst. Des. Anal. 3, 165–174 (2012) 3. Brinker, J.: Optimal design of functionally extended parallel robots with Delta-like architecture. Dissertation, RWTH Aachen University (2018) 4. Brinker, J., Corves, B., Takeda, Y.: Kinematic performance evaluation of highspeed Delta parallel robots based on motion/force transmission indices. Mech. Mach. Theory 125, 111–125 (2018) 5. Brinker, J., Schmitz, M., Takeda, Y., Corves, B.: Dynamic modeling of functionally extended delta-like parallel robots with virtual tree structures. In: ROMANSY 22 Robot design, dynamics and control, vol. 584, pp. 171–179 (2019) 6. Carbone, G.: Stiffness performance of multibody robotic systems. In: International Conference on Automation, Quality and Testing, Robotics, pp. 219–224. IEEE (2006) 7. Carbone, G., Ceccarelli, M.: Comparison of indices for stiffness performance evaluation. Front. Mech. Eng. China 5(3), 270–278 (2010) 8. Ceccarelli, M., Carbone, G.: A stiffness analysis for CaPaMan (Cassino parallel manipulator). Mech. Mach. Theory 37(5), 427–439 (2002) 9. Corves, B., Mirz, C., Brinker, J., Matsuura, D., Takeda, Y.: Stiffness analysis of delta parallel robots combining the virtual joint method with an FEA stiffness model. In: ROMANSY 22 Robot Design, Dynamics and Control, vol. 584, pp. 347–354 (2019) 10. Courteille, E., Deblaise, D., Maurine, P.: Design optimization of a delta-like parallel robot through global stiffness performance evaluation. In: International Conference on Intelligent Robots and Systems, pp. 5159–5166. IEEE (2009) 11. Gauthier, J.F., Angeles, J., Nokleby, S.: Optimization of a test trajectory for SCARA systems. In: Advances in Robot Kinematics, vol. 18, pp. 225–234. Springer (2008) 12. Gosselin, C.: Stiffness mapping for parallel manipulators. IEEE Trans. Robot. Autom. 6(3), 377–382 (1990) 13. Griffis, M., Duffy, J.: Kinestatic control: a novel theory for simultaneously regulating force and displacement. J. Mech. Des. 113, 508–515 (1991) 14. Kim, J., Park, C., Kim, J., Park, F.C.: Performance analysis of parallel mechanism architectures for CNC machining applications. J. Manuf. Sci. Eng. 122(4), 753–759 (2000)

404

C. Mirz et al.

15. Zhang, L., Song, Y.: Optimal design of the delta robot based on dynamics. In: International Conference on Robotics and Automation, pp. 336–341. IEEE, May 2011 16. Patterson, T., Lipkin, H.: Structure of robot compliance. J. Mech. Des. 115(3), 576–580 (1993) 17. Rosyid, A., El-Khasawneh, B., Alazzam, A.: Kinematic performance measures and optimization of parallel kinematics manipulators: a brief review. In: Hurtado, E.G. (ed.) Kinematics. InTech (2017) 18. Zou, T., Angeles, J.: The decoupling of the Cartesian stiffness matrix in the design of microaccelerometers. Multibody Syst. Dyn. 34(1), 1–21 (2015)

Singularity Free Mode Changes of a Redundantly Driven Two Limbs Six-Dof Parallel Robot Takashi Harada(B)

and Yuta Kunishige

Kindai University, Higashiosaka, Osaka 577-8502, Japan [email protected] https://sites.google.com/view/parallelmech/en

Abstract. A novel singularity free mode changes of a redundantly driven two limbs six-dof parallel robot ATARIGI is proposed in this paper. ATARIGI uses a total of eight actuators to drive a seven-dof mechanism, six-dof output node plus one-dof built-in differential screw. The parallel robot has multiple solutions to the inverse and forward kinematics. By changing postures from one solution to another, the parallel robot can expand its workspace. For ATARIGI, the bending direction of the elbow (over and under) of each limb and direction of the gripper (up and down) can be switched by the mode changes. Singularity and redundancy of ATARIGI are discussed by two Jacobian matrices which relate the velocity of the actuators and the mechanism. Conditions that ATARIGI loses redundancy but keeps the non-singularity are derived. Using the conditions, singularity free mode changes between “the elbows-over and gripper-down, suitable posture for the bottom working area,” and “the elbows-under and gripper-up, suitable for the upper working area” is proposed. Keywords: Parallel robot

1

· Mode change · Singularity · Redundancy

Introduction

The parallel robot has excellent mechanical characteristics such as high accuracy, high rigidity, and a small mass of the moving part [1]. However, mechanical interferences and singular configurations restrict its rotational and translational workspace. To expand the rotational workspace of the parallel robot, mechanical gimmicks such as crank [2], pulley, rack and pinion, or screw [3] are embedded inside the output node as an additional rotational mechanism. The authors have proposed a novel six-dof parallel robot ATARIGI that drives an output node with an embedded differential screw mechanism using two limbs redundantly driven by eight actuators [4,5]. The parallel robot has multiple solutions to the inverse and forward kinematics. By changing postures from one solution to another, the parallel robot This work was supported by JSPS KAKENHI Grant Number JP18K04068. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 405–413, 2021. https://doi.org/10.1007/978-3-030-58380-4_49

406

T. Harada and Y. Kunishige

can expand its translational workspace [6]. However, conventional non-redundant parallel robot encounters the singularity during the mode change. The actuation redundancy is one of the best solutions for avoiding the singularity during the mode change [7]. In this paper, singularity free mode changes of ATARIGI by the actuation redundancy for expanding the workspace are proposed.

Fig. 1. Schematic view of ATARIGI, two limbs six-dof parallel robot redundantly driven by eight actuators.

2

Kinematics of ATARIGI

Figure 1 illustrates the schematic view of ATARIGI. ATARIGI has four actuators on each limb that control the displacement di1 and the angles θi2 , θi3 and θi4 . ATARIGI uses a total of eight actuators to drive a seven-dof mechanism (six-dof output node position xm , ym and zm , and posture αm , βm and γm + one-dof built-in differential screw rotation γs ). Angle γg of the gripper is redundantly given by the sum of the angle γm of the output node and the angle γs of the differential screw. This means that ATARIGI has the kinematic redundancy. By the kinematic redundancy, ATARIGI improves its manipulability or avoids collisions between the limbs [4]. The seven-dimensional generalized force fx of the mechanism is redundantly given by the eight-dimensional generalized force fq of the actuators. This means that ATARIGI has the actuation redundancy from the actuators force to the mechanism force as well as the kinematic redundancy from the mechanism displacement to the gripper displacement. By the actuation redundancy, the actuators generate the internal force of the mechanism, which removes clearances around the mechanical pairs [4]. Figure 2 illustrates the kinematic model of ATARIGI. The loop-closure equation of the i th limb of ATARIGI is derived as, di1 zi0 + ai4 xi4 + ai5 xi5 = ±b0 zb + p ∓ lw + b1 ki k1 = u, k2 = v

(1)

Singularity Free Mode Changes of a Redundantly Driven Two Limbs

407

As shown in Fig. 1, the three-dimensional vector p represents the position Op of the central point of the output node with respect to the base coordinate frame. u, v and w are the unit direction vectors of the output node. l represents the distance of the nut from the central point of the output node, which gives the angle γs of the differential screw. zij and xij represent the unit direction vector of the coordinate system attached to the arms of ATARIGI.

Fig. 2. The kinematic skeleton model of ATARIGI

The seven-dimensional vector x and the eight-dimensional vector q represent generalized displacements of the output link with additional rotational mechanism (differential screw) and actuators respectively, i.e., T  x = xm ym zm αm βm γm γs T  q = d11 θ12 θ13 θ14 d21 θ22 θ23 θ24

(2a) (2b)

Differentiating Eq. (1) with respect to time, then organizing expression in matrix form, one obtains [4], Jx x˙ = Jq q˙ (3) Jx and Jq represent 8 × 7 and 8 × 8 Jacobian matrices as follows; ⎡ T ⎤ x15 ((b1 u − lw) × x15 )T −xT 15 w ⎢ uT (u × (b0 zb + p))T 0 ⎥ ⎢ T ⎥ ⎡ ⎤ T ⎢0 w 0 ⎥ I3 0 0 ⎢ T ⎥ ⎢ ⎢0 ⎥ −vT 0 ⎥ ⎥ · ⎢ 0 Rω 0 ⎥ = Jxb Jb Jx = ⎢ T ⎢ x25 ((b1 v + lw) × x25 )T xT ⎥ ⎣ w ld ⎦ 25 ⎢ T ⎥ 0 0 ⎢ v (v × (−b0 zb + p))T ⎥ 0 2π ⎢ ⎥ ⎣ 0T −wT 0 ⎦ 0T uT 0

(4)

408

T. Harada and Y. Kunishige

Jq =



⎡

Jq1 0 , Jqi 0 Jq2

⎤ T xT i5 zi0 ai4 xi5 Jxi4 ⎢ zT zi0 di1 zT Jzi3 ⎥ i3 i0 ⎥ =⎢ ⎣ 0 ⎦ tT i Jzi3 T 0 w Jzi3

(5)

In Eq. (4), Jb represents non-singular coordinate transformation matrix from  T x˙ to p˙ ω l˙ . The closed-form solutions of the inverse displacement from the mechanism to the actuators, and the forward displacement analysis from the actuators to the mechanism have derived in our previous research [4].

3

Singularity and Redundancy of ATARIGI

The singularity and the redundancy of ATARIGI, the seven-dof mechanism redundantly driven by the eight actuators are analyzed in this section. Conventional non-redundant parallel robot encounters singularity during the mode change. Singularity free mode change of the redundantly driven parallel robot can be achieved by the condition that the robot loses the redundancy but keeps non-singularity [7]. 3.1

Actuator Redundancy and Singularity

The eight actuators of ATARIGI has actuator redundancy when rankJq = 8

(6)

By the symbolic formula manipulation using Mathematica, one obtains, detJq = detJq1 · detJq2 = (a14 sin θ12 (sin θ13 )2 sin θ15 ) · (−a24 sin θ22 (sin θ23 )2 sin θ25 )

(7)

If one of the angles θ12 , θ13 , θ15 , θ22 , θ23 and θ25 equals nπ, then detJq equals zero, thus ATARIGI loses the actuation redundancy. Ranks of Jq in accordance with combinations of the angles are derived by Mathematica as shown in Table 1. The configurations of ATARIGI at the critical conditions of (a) θi2 = nπ, (b) θi2 = nπ and (a) θi5 = nπ are illustrated in Fig. 3(a)–(c). ATARIGI becomes the actuator singularity when rankJq = 6 At that time, ATARIGI cannot control the seven-dof mechanism.

(8)

Singularity Free Mode Changes of a Redundantly Driven Two Limbs

409

ATARIGI loses the actuation redundancy but keeps the actuator nonsingularity when (9) rankJq = 7 At that time, ATARIGI loses the actuation redundancy but can control the seven-dof mechanism. Table 1. Relationship between the angles of ATARIGI and rankJq Condition rankJq = 8 rankJq = 7 8 7a 7b 7c θ12 θ13 θ15 θ22 θ23 θ25

∗

—∗ — — — — —

= nπ — — — — —

— — = nπ — — —

— — — = nπ — —

7d

7e

7f

rankJq = 6 6a 6b 6c

6d

6e

6f

— — — — — = nπ

= nπ — = nπ — — —

— — — = nπ — = nπ

— = nπ — — — —

— — = nπ — — = nπ

= nπ — — — — = nπ

— — = nπ = nπ — —

— — — — = nπ —

= nπ — — = nπ — —

—represents = nπ

Fig. 3. Kinematic critical postures of ATARIGI.

3.2

End-Effector Singularity

The determinants of 7 × 7 minor matrices Jxbi which drops i th row of the 8 × 7 Jacobian matrix Jxb are given as, detJxb1 = detJxb2 = detJxb5 = detJxb6 = 0 T detJxb3 = detJxb4 = detJxb7 = detJxb8 = −(xT 15 w)(x25 w)

(10)

410

T. Harada and Y. Kunishige

When x15 ⊥ w or x25 ⊥ w, ATARIGI encounters the end-effector singularity. At that time, ATARIGI cannot control the seven-dof mechanism. The configuration of ATARIGI at the critical conditions of xi5 ⊥ w is illustrated in Fig. 3(d).

Fig. 4. Mode changes between (a) elbows-over and gripper-down and (b) elbows-under and gripper-up.

4

Enlargement the Workspace by Singularity Free Mode Changes

ATARIGI can change its postures between (a) elbows-over and gripper-down and (b) elbows-under and gripper-up as shown in Fig. 4 via the mode changes. By the mode changes as shown in Fig. 4, ATARIGI can expand its workspace not only the bottom area but also the upper area. The posture of the (a) elbows-over and gripper-down is suitable for the workspace of the bottom area, and the posture of the (b) elbows-under and gripper-up is suitable for the workspace of the upper area. In this section, singularity free mode changes from the (a) elbows-over and gripper-down to the (b) elbows-under and gripper-up are proposed. 4.1

Singularity-Free Working Mode Change Passing xi4  xi5

Figure 5 illustrates the working mode change passing x24  x25 . By directory controlling the actuated angle θ24 as shown in the figure, ATARIGI changes the limb 2’s elbow between over and under. For the working mode change, the forward displacement based actuator control must be applied, which are generally difficult for the parallel robot. The closed-form forward displacement solution of ATARIGI in the previous research solves this problem. During the mode change, ATARIGI encounters the critical posture of x24  x25 (θ25 = nπ) as shown in Fig. 5(b). At this posture, rank Jq is degenerated from 8 to 7, which corresponds to the condition 7d in Table 1. At that time, ATARIGI loses the redundancy but keeps the non-singularity. Note here that at this posture, other critical conditions such as x14  x15 (θ15 = nπ) of the condition 6d in Table 1, nor xi5 ⊥ w must not be simultaneously occurred for avoiding the singularity.

Singularity Free Mode Changes of a Redundantly Driven Two Limbs

411

Fig. 5. Singularity-free working mode change passing x24  x25 .

4.2

Singularity-Free Working Mode Change Passing zi0 ⊥ zi3

Figure 6 illustrates the working mode change passing z10 ⊥ z13 . By directory controlling the actuated angle θ12 as shown in the figure, ATARIGI changes the limb 1’s elbow between over and under, at that time the direction of the gripper is simultaneously changed between down and up. During the mode change, ATARIGI encounters the critical posture of z10 ⊥ z13 (θ12 = nπ) as shown in Fig. 6(b). At this posture, rank Jq is degenerated from 8 to 7 which corresponds to the condition 7a in Table 1. At that time, ATARIGI loses the redundancy

Fig. 6. Singularity-free working mode change passing zi0 ⊥ zi3 .

412

T. Harada and Y. Kunishige

but keeps the non-singularity. At the posture, other critical conditions such as x24  x25 (θ25 = nπ) of the condition 6e, nor xi5 ⊥ w must not be simultaneously occurred for avoiding the singularity. 4.3

Singularity Free Mode Changes Between Elbows-Over, Gripper-Down and Elbows-Under, Gripper-Up

By the combination of the singularity free working mode changes in the previous sub sections, ATARIGI switches its posture between the (a) elbows-over, gripperdown and the (b) elbows-under, gripper-up as shown in Fig. 7. ATARIGI expand its operating space both bottom and upper areas. Note here that the position of the linear axis of d11 is actively controlled during the mode changes between (b) and (d) in Fig. 7 for avoiding the critical condition of x25 ⊥ w.

Fig. 7. Singularity free mode changes between elbows-over, gripper-down and elbowsunder, gripper-up.

5

Conclusion

A novel singularity free mode changes of a redundantly driven two limbs six-dof parallel robot ATARIGI was proposed in this paper. Conditions that ATARIGI loses redundancy but keeps the non-singularity were derived. Using the conditions, singularity free mode changes between (a) the elbows-over and gripperdown, which is a suitable posture for the bottom working area, and (b) the elbows-under and gripper-up suitable for the upper working area was proposed.

Singularity Free Mode Changes of a Redundantly Driven Two Limbs

413

References 1. Merlet, J.: Parallel Robots. Springer, Dordrecht (2006) 2. Gauthier, J., Angeles, J., Nokleby, S.B., Morozov, A.: The kinetostatic conditioning of two-limb Sch¨ onflies motion generators. Trans. ASME J. Mech. Robot. 1, 0110101–12 (2009). https://doi.org/10.1115/1.2960544 3. Company, O., Pierrot, F., Krut, S., Nabat, V.: Simplified dynamic modeling and improvement of a four-degree-of-freedom pick-and-place manipulator with articulated moving platform. Proc. Inst. Mech. Eng. 223(1), 13–29 (2009) 4. Harada, T., Angeles, J.: From the McGill pepper-mill carrier to the Kindai ATARIGI carrier, a novel two limbs six-dof parallel robot with kinematic and actuation redundancy, In: Proceedings of the 2017 IEEE ROBIO, Macau, pp. 1328–1333 (2017). https://doi.org/10.1109/ROBIO.2017.8324601 5. Harada, T., Makino, T.: Closed-form solutions of the kinetostatics of a two-limbs six-dof parallel mechanism with kinematic and actuation redundancies. Trans. JSME 84(866), 18–00025 (2018). https://doi.org/10.1299/transjsme.18-00025. (in Japanese) 6. Harada, T.: How to expand the workspace of parallel robots (chap. 5). In: Kinematics, pp. 95– 111. InTech (2017). https://doi.org/10.5772/intechopen.71407 7. Harada, T.: Mode changes of redundantly actuated asymmetric parallel mechanism. J. Mech. Eng. Sci. 230(3), 452–464 (2015). https://doi.org/10.1177/ 0954406215588479

Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory Arati Ajay Bhattu(B) and Salil Kulkarni Indian Institute of Technology Bombay, Powai 400076, India [email protected]

Abstract. Continuum robots bio-inspired from elephant trunk or snake are widely researched due to its curvilinear path following capabilities in restricted environment. Tendon actuation is one of the direct approaches to bend such continuous structures with high force generation ability. This paper focuses on developing kinematic and dynamic model for tendon actuated robot based on cosserat rod theory with piecewise constant curvature method. Representing rotational configuration of such robots using quaternions instead of traditional Euler angles eliminates singularity in modeling and ensures unique positioning of end effector even for large relative rotations. The actuation model developed incorporates general loading on element with piecewise constant curvature and is applicable for robots with any number of discs, tendons and configuration of tendons. This model accurately predicts bending, torsion, shear and extension of robot which is validated with standard benchmark problems of two, three dimensional co-rotational as well as curved beams subjected to nonlinear external load.

Keywords: Continuum robot

1

· Quaternions · Tendon actuation

Introduction

The continuum robots are inspired by octopus tentacles and elephant trunks with capability of continuous elastic deformation. They have wide applications in medical [1,2], nuclear [3] and space technologies [4]. Hence, there is a need to develop an accurate and numerically efficient dynamic model of such robots. There are three kinematic frameworks to model continuum robots viz. discrete rigid links [5], constant curvature [6] and variable curvature [7]. In other cases, these models have been classified as lumped parameter models [8,9] and distributed parameter models [10]. In lumped parameter model, continuum body is assumed to be collection finite number of mass and springs. Whereas in distributed parameter model, Hamilton’s principle is used to derive PDEs under pure bending. In addition, there are two mechanical frameworks namely; energy method and force equilibrium to model the external forces like tendon actuations. The tendon actuation models in the literature use the bending plane at the point of actuation as a reference to calculate the actuation forces [11,12]. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 414–422, 2021. https://doi.org/10.1007/978-3-030-58380-4_50

Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory

415

In this paper, quaternions are used in place of conventional Euler angles to avoid the modifications in intrinsic curvatures and bending energies as a result of singularities associated with certain turning angles [13]. The piecewise constant curvature approach from cosserat rod is implemented with flexibility of both energy and force methods to add variety of external actuations. Steps of development of energy based rod model and its use in tendon actuated robot are explained in subsequent sections. The equation formulation consists of internal forces (Sect. 2) and external forces due to gravity and tendon actuation (Sect. 3). This model is validated with benchmark beam problems and then applied to the tendon actuated robots (Sect. 4).

2 2.1

Dynamic Modeling by Rod Theory Kinematics of Rod

A general representation of Cosserat rod is explained in this section. The configuration of rod is defined by r(σ) which denotes the position of the centreline of the rod. Here, σ is the dimensionless parameter varying from 0 to 1 along the rod. For any given σ, orthogonal coordinate system has been defined as di (σ)∀i = 1, 2, 3 where, di are the components of the orthonormal basis vector [14]. From differential geometry, there exist vectors u and ω such that, di ∗ = u×di , d˙i = ω × di . Here, di ∗ and d˙i represent spatial and temporal derivative of di . These vectors are called Darboux vectors. Calculation of Darboux vector involves the concept of rotation matrix which is explained in Sect. 2.2. 2.2

Rotation Representation

To capture orientation of directors (di ) with respect to global frame of reference, an orthogonal rotation matrix R is defined. To avoid Gimbal lock conditions, quaternions are used instead of conventional Euler angles. The quaternions are defined as q = [q1 q2 q3 q4 ]. The sum of square of four quaternions is assumed to be unity. This rotation matrix can orient any body fixed vector to global coordinates as [15],  (1) r = Rr 

Here, r and r are the position vectors in body fixed frame and global reference frame with same origin. Henceforth, any parameter k is defined as k  in body fixed frame. R = R(q) where, set of quaternions is defined in the relative direc tion of r and r . One rotation matrix corresponds to unique set of quaternions. This matrix is useful for calculating strains which is explained in Sect. 2.3. 2.3

Strain Calculations

This section explains the calculation of bending strain in rod theory. The bending strains are obtained by taking time derivative of Eq. (1). This is further simplified

416

A. A. Bhattu and S. Kulkarni

 on  the basis of definition of body fixed vectors and orthogonality of R as, r˙ = ˙ r = ω × r . Here, RT R ˙ is a skew symmetric matrix. From definition RT R of Darboux vector stated in Sect. 2.1, components strain and angular velocity expressions are obtained.

2.4

Energy in Continuous Form

Potential Energies. Potential energy due to bending and torsion is given by [16],  T   1 1  u(σ) − uk (σ) Ck (σ) u(σ) − uk (σ) dσ Ebt = (2) 2 0 where, Ck is a diagonal matrix with flexural rigidity in x, y, z direction. Similarly, potential energy associated with shear and stretching for given strain ψ and diagonal rigidity matrix Cp is, Ess

1 = 2



1

ψ(σ)T Cp ψ(σ)dσ

(3)

0

Kinetic Energies. Rotational   and Ttranslational kinetic energies are Erk = 1 1 1 1 T ω(σ) I ω(σ) dσ, E = tk 2 0 2 0 ρ A r rdσ. Here, ω, I and ρ are angular velocity, inertia tensor and density respectively. Similarly, for each discrete element energy equations are written based on linear interpolation of parameters. 2.5

Combined Lagrange-Euler Approach

Lagrange Equation when applied to translational energy of a discrete element, simplified equations are obtained in the form of F = ma . However, complex inertial terms are generated in the case of rotational energy due to use of quaternions. Lagrange and Euler equations of motion for rotation in body fixed frame are,  ∂L d ∂L + Qp = I ω˙ + ω  × I ω  = τ  (4)



 dt ∂ p˙ ∂p



 Inertia terms Moments Inertia terms

Torque parameters

L is the difference between kinetic and potential energies. Here p is a generalized coordinate (quaternions) and Qp is the corresponding generalized force. ω  and I are angular velocity vector in body fixed frame and inertia tensor respectively. To avoid the complexity in inertia terms in Lagrange equation and still use the benefits of energy approach, both Lagrange and Euler equations are combined using transformation matrix (Q) and written for discrete element i as,      Ii ω˙i + ω i × Ii ωi 1 ¯ T ∂L = Qi + Qpi (5) 2 ∂pi 0

Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory

417

These equations are solved numerically with Newton-Raphson method along with Newmark Beta time integration scheme (β = 14 and γ = 12 ). The equation formulation from this section is used to model external tendon actuations which is explained in the next section.

3

Tendon Actuation Modeling

The tendons can generate relatively high forces with extrinsic actuation. The tendons can be arranged in a straight, cross, or helical way. Among these, straight tendon is presented here. As shown in Fig. 2, the tendons are passed through support discs mounted on elastic rod. Here, the interaction between disc and tendon is assumed to be frictionless. Hence, tension remains constant throughout the string.

Fig. 1. Tendon actuated robot

Fig. 2. String robot [18]

In this continuum robot, the actuation moments and forces are due to string tensions. The tension is acting in the direction of adjacent disc-hole directions. Free body diagram of disc is shown in Fig. 3. In global coordinates, hole position is Ri = R + Rot ri . Where, Rot is rotation matrix associated with ri and R is the position vector of rod centre at disc i. For adjacent discs, same notation is used with index i+1 and i-1.

Fig. 3. FBD

Fig. 4. Position of holes

Fig. 5. Ramp actuation

418

A. A. Bhattu and S. Kulkarni

Force on discrete rod element at disc i in global coordinates and moment in local coordinate system due to string tension T is,     Ri−1 − Ri Ri+1 − Ri +T , MT  = ri × Rot−1 FT (6) FT = T ||Ri+1 − Ri || ||Ri−1 − Ri || Uniformly Distributed Forces Due to Gravity. Gravitational force vector F rj = mj g, is added on every discrete element j , where mj is the lumped mass of each discrete element j. The disc inertia is also considered along with rod. The model developed is applied to some benchmark problems and continuum robot in the next section.

4 4.1

Model Validation Benchmark Problems

To validate the model developed, some benchmark problems of two and threedimensional co-rotational beams are selected. Here, the external forces due to tendon and gravity are replaced by the given external load. Along with straight beam, curved beam problem is also validated. Two problems are presented here (Fig. 7).

Fig. 6. Geometric parameters of cantilever beam [17]

Fig. 7. Force input

A problem (Fig. 6) is taken from a paper [17] related to co-rotational two dimensional beams undergoing large deformations. The loading, material and

Fig. 8. Validation of results (v)

Fig. 9. Validation of results (u)

Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory

419

geometric conditions are given in Section 6.1 of the same paper. The amplitude of sinusoidal force is high enough to cause large displacement of beam. Plots of transverse displacement and axial displacement are shown in Fig. 8 and 9. Reference solution has 48 elements and new formulation has 20. The reference solution from paper has been checked using Total Lagrangian formulation. Similarly, a curved beam problem is validated from paper [17]. The geometric conditions, loading and material descriptions are taken from Section 6.2. Figure 11 shows vertical displacement history at the point of application of force. There are 48 elements in reference solution and 40 in new formulation (Fig. 10).

Fig. 10. Shallow cantilever beam [17]

4.2

Fig. 11. Validation of results

Simulation Results of Continuum Robot

In this section, the dynamic response of continuum robot (Figure 1) with 11 discs (including fixed disc) is simulated. The geometric and material properties listed in Table 1 are taken from [19]. Table 1. Material and geometric properties Elastic modulus (E) 8.75.107 Pa Poisson’s ratio (ν) 3

0.3

Density (ρ)

850 kg/m

Radius of core (a)

0.01 m

Radius of disc (Rd )

0.05 m

Disc spacing (s)

0.1 m

length of disc (ld )

5 mm

Number of elements 20

Actuation at Position 1. Here, it is assumed that the undeformed rod is oriented along Z direction. We add a string at position 1 of every disc (Fig. 4) and apply ramped tension input as shown in the Fig. 5. In this case, rod reaches extreme end in negative Y direction due to gravity in 0.63 s. After that, rod starts moving in positive Y direction. The Fig. 12 shows snap of rod from 0.63 s to 1.42 s. Red dots represent the position of disc. Figure 13 shows the graph when tension remains constant. The rod again reaches local minimum at 2.13 s and then starts moving upward.

420

A. A. Bhattu and S. Kulkarni

Fig. 12. Plots for time 0.6 to 1.4 s

Fig. 13. Plots for time 2.2 to 2.9 s

Actuation at Position 1 and 2. In this case, uniformly increasing tension is applied at position 1 (T1) and 2 (T2). T1 increases from 0 to 20 N in 2 s and T2 increases from 0 to 10 N in 2 s. Figure 14 shows the Y-Z and X-Z plot of rod for time t = 0 to 0.6 s. Rod reaches extreme end of negative X direction in 0.91 s and reaches extreme end of negative Y direction in 0.60 s. The Cosserat rod model is validated using large deformation beams because the elastic core of continuum robot shows similar internal forces. The tendon actuation can be assumed as varying external load along beam sections.

Fig. 14. Plots for time 0 to 0.6 s

5

Conclusion

The model developed in this paper can accurately predict the behavior of straight tendon actuated robot undergoing large deformation due to bending, shear, torsion and elongation. The study can be extended to crossed and helical tendon driven robots. In addition, modeling of any continuum robot with external springs, drag or uniformly distributed loads with the consideration of string-disc friction and material damping effects is possible. Future scope of this work includes detailed validation of tendon actuation with literature and development of inverse kinematics of continuum robot which involves prediction of actuation forces for a predefined target position. With

Dynamics of Tendon Actuated Continuum Robots by Cosserat Rod Theory

421

the use of quaternions, unique mapping of angular position, rotation matrix and actuation force is possible thereby paving way for development of control strategies for continuum robots.

References 1. Lau, K., Leung, Y., Yam, Y.: Applications of flexible robots in endoscopic surgery. Handbook of Robotic and Image-Guided Surgery, pp. 303–322 (2020). Elsevier 2. Russo, S.: Smart composites and hybrid soft-foldable technologies for minimally invasive surgical robots. Handbook of Robotic and Image-Guided Surgery, pp. 323– 340 (2020). Elsevier 3. Buckingham, R., Graham, A.: Nuclear Snake-ARM robots. Industrial Robot: An International Journal (2012) 4. Santiago, J.L., Walker, I.D., Godage, I.S.: Continuum robots for space applications based on layer-jamming scales with stiffening capability. IEEE Aerospace Conference, pp. 1–13 (2015) 5. Kutzer, M.D., Segreti, S.M.: Design of a new cable-driven manipulator with a large open lumen: preliminary applications in the minimally-invasive removal of osteolysis. IEEE International Conference on Robotics and Automation, pp. 2913– 2920 (2011) 6. Qu, T., Chen, J., Shen, S., Xiao, Z., Yue, Z., Lau, H.Y.: Motion control of a bioinspired wire-driven multi-backbone continuum minimally invasive surgical manipulator. IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 1989–1995 (2016) 7. Rone, W.S., Ben-Tzvi, P.: Mechanics modeling of multisegment rod-driven continuum robots. J. Mech. Robot. 2(4), 103 (2016) 8. Jung, J., Penning, R.S., Ferrier, N.J., Zinn, M.R.: A modeling approach for continuum robotic manipulators: effects of nonlinear internal device friction. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5139– 5146 (2011) 9. Falkenhahn, V., Mahl, T., Hildebrandt, A., Neumann, R., Sawodny, O.: Dynamic modeling of bellows-actuated continuum robots using the euler-lagrange formalism. IEEE Trans. Robot. 31(6), 1483–96 (2015) 10. Gravagne, I.A., Rahn, C.D., Walker, I.D.: Large deflection dynamics and control for planar continuum robots. IEEE/ASME Trans. Mechatron. 8(2), 299–307 (2003) 11. Rucker, D.C., Webster III, R.J.: Statics and dynamics of continuum robots with general tendon routing and external loading. In: IEEE Transactions on Robotics, pp. 1033-1044 (2011) 12. Dalvand, M.M., Nahavandi, S.: Howe RD. an analytical loading model for n- tendon continuum robots. IEEE Trans. Robot. 34(5), 1215–1225 (2018) 13. Goldberg, N.N., Huang, X., Majidi, C.: On Planar Discrete Elastic Rod Models for the Locomotion of Soft Robots. 6(5), 595–610 (2016) 14. Antman, S.S.: Nonlinear Problems of Elasticity, vol. 107. Springer (1995) 15. Spillmann, J., Teschner, M.: CoRdE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 63–72 (2007) 16. Lang, H., Linn, J., Arnold, M.: Multi-body dynamics simulation of geometrically exact cosserat rods. Multibody Syst. Dyn. 25(3), 285–312 (2011)

422

A. A. Bhattu and S. Kulkarni

17. Le, T.N., Battini, J.M., Hjiaj, M.: Efficient formulation for dynamics of corotational 2D beams. Computat. Mech. 48(2), 153–61 (2011) 18. Youtube. https://www.youtube.com/watch?v=EUEp-AfvvzE. Accessed 4 Aug 2015 19. Arbind, A., Reddy, J.N.: Transient analysis of cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model. Int. J. Non-Linear Mech. 79, 38–47 (2016)

Proposition of On-Line Velocity Scaling Algorithm for Task Space Trajectories Marek Wojtyra(B)

and L  ukasz Woli´ nski(B)

Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland {mwojtyra,lwolinski}@meil.pw.edu.pl

Abstract. A preliminary proposition of a trajectory scaling algorithm is presented. The new algorithm allows for planning a task space trajectory while taking into account the physical capabilities of the manipulator in terms of limits for joint space velocities and accelerations. Contrary to the classic scaling algorithm, which scales uniformly the whole trajectory and has to be executed before the motion starts, the proposed algorithm can be executed online and the scaling is applied only to the part of the planned trajectory. In the proposed algorithm, the planned motion is analyzed in a specified prediction horizon. Theoretical basics of the classic trajectory scaling method are adapted in such a way that the scaling commences only when the analysis detects a violation of joint velocity or acceleration constraints at the prediction horizon. In this paper, the underlying ideas are introduced and discussed, then the core formulae of the algorithms are detailed. An example of scaling a trajectory of a 2-DOF manipulator in the vicinity of a boundary singularity is presented.

Keywords: Trajectory scaling

1

· Robot manipulators · Kinematics

Introduction

Trajectory generation is one of the most essential topics in robotics, extensively studied and covered in the literature. A trajectory consists of a path in space and a time law specifying how fast that path should be traversed. There are many methods for trajectory generation, making use of polynomials, splines, trigonometric functions, and more [1]. Generating the trajectory in the joint space, a classic point-to-point motion, is pretty straightforward, as the physical capabilities of the robot usually translate to the constraints on the joint velocities and accelerations. On the other hand, the trajectory in the task space requires solving of the manipulator inverse kinematics problem. At this point, the solution might turn out to be not feasible because it exceeds the robot limits. Such a scenario is especially possible in case

Supported by the National Science Centre (Poland) grant 2018/29/B/ST8/00374. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 423–431, 2021. https://doi.org/10.1007/978-3-030-58380-4_51

424

M. Wojtyra and L  . Woli´ nski

when the whole trajectory is not known in advance but generated online. Consequently, the actuator saturation leads to the path deviation and unacceptable trajectory tracking errors. To this day, several online trajectory generation algorithms were developed [2,9,11]. Their main domain is the joint space. They could be used to generate task space trajectories, however, translating joint limits into constraints on the task space is not a solved problem. In [3,10], methods for dynamic scaling of generated trajectories are proposed. The algorithms for scaling the trajectories subject to kinematic or dynamic constraints are further developed in [5,7,8]. However, these methods might need special techniques to preserve their stability [6]. All of the previously cited methods compute the trajectory only for the next step and do not use any prediction horizon. Therefore, the joint limits could still be exceeded. Taking into account future points of the trajectory would solve that problem [4,12–14]. In this framework, we propose an algorithm for task trajectory scaling, which checks the satisfaction of the joint constraints in a specified prediction horizon. Note that the traveled path does not change due to scaling, thus its important properties, e.g., obstacle collision avoidance, remain unchanged. This paper is organized as follows: Sect. 2 describes the classic trajectory scaling method, Sect. 3 proposes a new algorithm, Sect. 4 deals with the numerical example showing the trajectory scaling task for a planar robot, and Sect. 5 concludes the paper.

2

Trajectory Scaling

In this paper, trajectories in the Cartesian space are considered. A trajectory can be defined as a path p in spacetime: p(t) = p(s(τ )),

(1)

where s(τ ) is a path parameter dependent on τ , which is time scaled to the duration of motion T : τ = Tt . The path p(s) can describe any curve in the Cartesian space, and s(τ ) can be any continuous, increasing, differentiable function of time. In addition, a time derivative of the path parameter s(τ ˙ ) represents a velocity profile, for example a trapezoidal profile. In the classic method [1], first, the inverse kinematics problem is solved for the whole trajectory. Then, the joint velocity and acceleration limits are being checked separately. The coefficient λv is the measure of how much the joint velocity limit is exceeded by the most demanding joint:   |q˙j | |q˙1 | |q˙n | , . . . , vmax,j , . . . , vmax,n , (2) λv = max 1, vmax,1 where |q˙j | is the maximum j-th joint velocity and vmax,j is the j-th joint velocity limit.

Proposition of On-Line Velocity Scaling Algorithm

425

The coefficient λa is the measure of how much the joint acceleration limit is exceeded by the most demanding joint:   |¨ qj | |¨ q1 | |¨ qn | , . . . , amax,j , . . . , amax,n λa = max 1, amax,1 , (3) where |¨ qj | is the maximum j-th joint acceleration and vmax,j is the j-th joint acceleration limit. Finally, the coefficient λ is computed as the greater value of λv and λa :  √  (4) λ = max λv , λa and becomes the factor which scales the duration of the motion T ∗ = λT . Correspondingly, the scaled time becomes t∗ = λt while the joint velocites are ˙ being scaled by dividing by λ: q˙ ∗ (t∗ ) = q(t) λ , and joint accelerations by dividing ¨ q (t) ∗ 2 ¨ ∗ by λ : q (t ) = λ2 . As the number of calls of the inverse kinematics solver grows in pair with the number of considered points on the trajectory, the classic method is rather suited for offline use, before the manipulator motion could start.

3 3.1

Proposed Algorithm Underlying Ideas

The classic method slows down motion along the whole path segment and all calculations must be done prior to the execution of the planned motion. Contrary to the classic method, the algorithm proposed here can be realized online, i.e., during the execution of the desired trajectory. Moreover, motion slows down only along a part of the path. The basic idea of the proposed algorithm consists in scaling the trajectory, similarly to what is done in the classic method, however, the scaling is not performed for the entire trajectory segment. Slowing down the planned motion commences only when it is necessary. To do so, the planned motion is analyzed in a specified prediction horizon, several (namely N ) control steps ahead of the current time. For this prediction horizon, the inverse kinematic problem is solved to check whether or not the limits for joint velocities and accelerations are exceeded. If limits are violated, then a scale factor (as of Eq. (4)) is calculated and used to slow down the originally planned motion. The procedure of slowing down the motion—i.e., of adjusting the originally assumed velocity profile—is applied to all points of the planned trajectory between the current time and the prediction horizon (points of the trajectory correspond to the subsequent time steps of the control system). Figuratively, the procedure for adjusting the speed profile can be compared to driving a car—to stay on the road, the driver must start braking before turning. To explain the details of the proposed algorithm, several concepts shall be introduced and discussed.

426

M. Wojtyra and L  . Woli´ nski

Firstly, it is important to know that trajectory points between the current time tcur (for the j-th control step tcur = jdt) and the prediction horizon thor = tcur + N dt are calculated in advance and stored in a memory buffer. When necessary, i.e., when the trajectory must be scaled to meet the imposed limits, the values stored in the buffer are being modified. Secondly, it should be noted that—at each control step—if the algorithm works properly, the trajectory points previously placed in the buffer fulfill the imposed limits. That is why it is enough to check the fulfillment of the limits only for the new point which is to be placed at the end of the memory buffer. For this point, scale factor λj is calculated using Eq. (4). Thirdly, function s(τ ) shall be discussed. Parameter s serves as a motion advancement measure—it equals 0 at the beginning of the path and equals 1 at the path end. In the classic approach parameter τ is the normalized time; when calculating the trajectory, the robot control system samples τ at regular intervals dt, the same for the whole trajectory, so that the increment of τ is always dτ = dt T . In the proposed algorithm, however, the scaling of the trajectory is realized by adjusting dτ . Thus, the trajectory points kept in the memory buffer, i.e. the points between the current time tcur (which corresponds to τcur ) and the prediction horizon thor are calculated for the following values of τ : τi = τcur + idτ = τcur + i

1 dt 1 dt = τi−1 + , i = 1, . . . , N. σj T σj T

(5)

The cumulative scale factor σj is originally set to 1 and then, at each control step j, is multiplied by the current scale factor λj : σj = σj−1 λj . It should be stressed that, if the scale factor λj is greater than 1 (if trajectory scaling is needed), then all values in the memory buffer are recalculated. On the other hand, if the scale factor λj equals 1 (which means: in the j-th step no further scaling is needed), then Eq. (5) does not have to be used for the points stored in the memory buffer—only a new point must be added at the buffer end. Finally, as soon as the first value stored in the memory buffer was utilized by the robot controller to execute motion, the values kept in the buffer are shifted to the left and calculations are repeated for the next control step. 3.2

Detailed Algorithm

For the proposed trajectory scaling algorithm, it is crucial to check the limits for joint velocity and acceleration and to calculate the scale factor. An auxiliary procedure Limits is used to find the scale factor λ for the given motion advancement measure si (λ = Limits(si )). The following calculations are performed: – Generate a point of the trajectory in the Cartesian space: pi = T rajGen(si ). – Solve the inverse kinematics problem: qi = InvKin(pi ).  T ¨i – Calculate joint velocities and accelerations: q˙ i q = V elAcc(qi , qi−1 , qi−2 ). ¨ i into Eqs. (2) and (3), – Calculate the scale factor λ—substitute q˙ i and q respectively, then use Eq. (4).

Proposition of On-Line Velocity Scaling Algorithm

427

In the algorithm proposed here, the consecutive values of motion advancement parameters si , for all time steps in the prediction horizon, are stored in the memory buffer sBuf . Note that in procedure Limits, to estimate velocities and accelerations, preceding values of q are used (backward finite difference formulae are utilized). Prior values of s (si−1 , si−2 , etc.) retrieved from sBuf can be used to calculate qi−1 and qi−2 . Since the values of s stored in the buffer change during the scaling process, there is no point in storing q. The calculations in the proposed here on-line trajectory scaling algorithm are organized as follows.

Algorithm 1. Trajectory scaling algorithm (Initialization) for i = 0, . . . , N− 1do sBuf (i) = s i dt T end for σ = 1, τcur = 0, τhor = (N − 1) dt T (Main loop) while τhor <  1 do dt  dt , λ = Limits(sN ), τ1 = τcur + σT , σ = σλ sN = s τhor + σT if λ > 1 then for i = 2, . . . , N do dt , sBuf (i) = s(τi ) τi = τi−1 + σT end for else   dt sBuf (N ) = s τhor + σT end if for i = 1, . . . , N do sBuf (i − 1) = sBuf (i) end for send sBuf (1) to the robot position controller (execute motion) dt , τcur = τ1 τhor = τcur + N σT end while for i = 2, . . . , N do send sBuf (i) to the robot position controller (execute motion) end for

For the sake of simplicity, in the presented algorithm it was assumed that scaling of the trajectory is not needed during the first N − 1 steps.

4

Numerical Example

The proposed algorithm was tested on a model of a 2-DOF planar manipulator. The lengths of the links were chosen as l1 = 1 m and l2 = 0.5 m. The kinematic  T  T 1, 2 rad limits of the actuators were vmax = 1, 1.5 rad s and amax = s2 . The desired end effector trajectory was generated as motion along a straight line from

428

M. Wojtyra and L  . Woli´ nski

 T  T PA = −1, 0.505 m to PB = 1, 0.505 m with trapezoidal velocity profile with nominal time T = 5 s and acceleration and deceleration periods both equal to 0.2 T . The number of prediction steps was set to N = 20. The manipulator in its initial and final configurations is shown in Fig. 1), along with the linear path and the workspace limits (represented by circular arcs). 1.5

y

1

0.5

0

-1.5

-1

-0.5

0

0.5

1

1.5

x

Fig. 1. A 2DOF planar manipulator.

The desired trajectory is too demanding and violates the velocity and acceleration limits. Application of the classic method, briefly summarized in Sect. 2, slows down the motion and extends its duration to 11 s. Meanwhile, the proposed algorithm from Sect. 3 results in the motion duration of 8.19 s. s(t)

1

ds(t)

0.3 0.2

0.5 0.1 0

0 0

4

8

12

0

4

8

12

Fig. 2. Path parameter s(t) (left) and path velocity s(t) ˙ (right). The classic method is in dashed magenta, and the proposed algorithm is in solid blue.

In Fig. 2 time evolution of the path parameter s(t) and velocity s(t) ˙ are shown. The desired trapezoidal velocity profile is tracked up to 2.22 s when the algorithm slows down the motion to avoid exceeding the joint limits. It is worth noting that the velocity is gradually slowed down to the level of the classic method. Consequently, the path parameter evolution of the proposed algorithm becomes parallel to that of the classic method around the 2.22 s mark, but still finishes faster.

Proposition of On-Line Velocity Scaling Algorithm

429

The path acceleration evolution s¨(t) is shown in Fig. 3. A max-zero-min profile is modified around 2.22 s due to the slowdown. The scaling parameter λ(t) (also in Fig. 3) starts changing its value from λ = 1 to finally reach λ = 2.201. Joint velocities q˙1 (t) and q˙2 (t) seen in Fig. 4 for both the proposed algorithm and the classic method are safely within their limits (shown as red lines). d2s(t)

0.5

scale (t)

2.5

0

2

-0.5

1.5

-1

1 0

4

8

12

0

4

8

12

Fig. 3. Path acceleration s¨(t) (left) and scaling parameter λ(t) (right). The classic method is in dashed magenta, and the proposed algorithm is in solid blue.

dq1(t)

1

dq2(t)

2

0

0

-1

-2 0

4

8

12

0

4

8

12

Fig. 4. Joint velocity q˙1 (t) (left) and q˙2 (t) (right), in rad . The classic method is in s dashed magenta, and the proposed algorithm is in solid blue.

d2q1(t)

1

d2q2(t)

2

0

0

-1

-2 0

4

8

12

0

4

8

12

Fig. 5. Joint acceleration q¨1 (t) (left) and q¨2 (t) (right), in rad . The classic method is in s2 dashed magenta, and the proposed algorithm is in solid blue.

In Fig. 5 joint accelerations q¨1 (t) and q¨2 (t) for the proposed algorithm and the classic method are pictured. For a brief moment, accelerations only touch their limits (shown as red lines) without exceeding them.

430

5

M. Wojtyra and L  . Woli´ nski

Final Remarks and Conclusions

A new algorithm for trajectory scaling was proposed and tested on a numerical example which illustrated its advantages over the classic method. In the classic method the whole trajectory is scaled uniformly (and only once). This guarantees that the joint limits for velocities and accelerations are not violated. In the proposed method, however, the trajectory is scaled partially. Moreover, within the prediction horizon (within the memory buffer), the same point may be scaled several times. As a result, when the cumulative scale factor σ changes, at the transition point between the previously scaled (or not yet scaled) trajectory and the newly scaled trajectory the required motion deceleration may be large. Consequently, the joint limits for velocities and accelerations may be violated. In the modified version of the algorithm presented here, such problems may be detected and alleviated, at the expense of additional calculations. Due to paper length restrictions, these issues are omitted here. A natural continuation of the proposed algorithm is to speed up the motion back to the originally planned velocity as soon as following the desired trajectory does not violate the limits. This part of the algorithm is more complicated and computationally complex; it will be presented in a separate publication. A natural extension of the algorithm is devising a method for choosing the best number of the prediction steps N , which represents the length of the horizon. It is planned for near future.

References 1. Biagiotti, L., Melchiorri, C.: Trajectory Planning for Automatic Machines and Robots. Springer, Heidelberg (2008) 2. Broquere, X., Sidobre, D., Herrera-Aguilar, I.: Soft motion trajectory planner for service manipulator robot. IROS 2008, 2808–2813 (2008) 3. Dahl, O., Nielsen, L.: Torque-limited path following by online trajectory time scaling. IEEE Trans. Robot. Autom. 6(5), 554–561 (1990) 4. Faroni, M., Beschi, M., Visioli, A.: Predictive inverse kinematics for redundant manipulators: Evaluation in re-planning scenarios. IFAC-PapersOnLine 55(22), 238–243 (2018) 5. Gerelli, O., Guarino Lo Bianco, C.: Nonlinear variable structure filter for the online trajectory scaling. IEEE Trans. Ind. Electron. 56(10), 3921–3930 (2009) 6. Guarino Lo Bianco, C., Ghilardelli, F.: Techniques to preserve the stability of a trajectory scaling algorithm. In: ICRA 2013, pp. 870–876 May 2013 7. Guarino Lo Bianco, C., Ghilardelli, F.: Real-time planner in the operational space for the automatic handling of kinematic constraints. IEEE Trans. Autom. Sci. Eng. 11(3), 730–739 (2014) 8. Guarino Lo Bianco, C., Wahl, F.M.: A novel second order filter for the real-time trajectory scaling. In: ICRA 2011, pp. 5813–5818 (2011) 9. Haschke, R., Weitnauer, E., Ritter, H.: On-line planning of time-optimal, jerklimited trajectories. IROS 2008, 3248–3253 (2008) 10. Hollerbach, J.M.: Dynamic scaling of manipulator trajectories. In: 1983 American Control Conference, pp. 752–756 (1983)

Proposition of On-Line Velocity Scaling Algorithm

431

11. Kr¨ oger, T., Tomiczek, A., Wahl, F.M.: Towards on-line trajectory computation. IROS 2006, 736–741 (2006) 12. Lange, F., Albu-Sch¨ affer, A.: Path-accurate online trajectory generation for jerklimited industrial robots. IEEE Robot. Autom. Lett. 1(1), 82–89 (2016) 13. Lange, F., Suppa, M.: Predictive path-accurate scaling of a sensor-based defined trajectory. ICRA 2014, 754–759 (2014) 14. Lange, F., Albu-Sch¨ affer, A.: Iterative path-accurate trajectory generation for fast sensor-based motion of robot arms. Adv. Robot. 30(21), 1380–1394 (2016)

Stability Analysis and Reconfiguration Strategy for Multi-agent D-Formation Control Alessandro Colotti1(B) , Angelo Cenedese1 , S´ebastien Briot2 , Isabelle Fantoni2 , and Alexandre Goldsztejn2 1

Department of Information Engineering, University of Padova, Padova, Italy [email protected], [email protected] 2 CNRS, Laboratoire des Sciences du Num´erique de Nantes, Nantes, France {Sebastien.Briot,Isabelle.Fantoni,Alexandre.Goldsztejn}@ls2n.fr

Abstract. This paper introduces a new control approach to perform formation control tasks on multi-agent systems, called D-formation control. The D-formation controller is a gradient-descent control law that exploits a regularized potential function to efficiently achieve specific formations. Taking inspiration from the flocking of birds, this approach differentiates itself from the several formation control strategies that can be found in the literature thanks to its flexibility. In fact, the approach that is usually employed in formation control is to try to enforce a set of very strict constraints in order to achieve rigid, a priori defined structures. We will show that the D-formation approach greatly relaxes such conditions. In this paper, the D-formation control problem is introduced, and the equilibrium configurations of the controller are characterized. Additionally, a strategy for switching from one stable equilibrium to another one, i.e. for changing the shape of the formation, is proposed. Keywords: Formation control

1

· Stability · Formation switching

Introduction

In the last few years, the control of multi-agent systems has sparked a significant amount of interest in research [10], due to both their practical employment in various applications [2,9] and the theoretical challenges arising in coordination and control of them [11,12]. A multi-agent system is a system formed by a network of mobile agents, usually aerial or terrestrial, which can interact and exchange information with each other in order to achieve a common goal. In particular, formation control [6], which is one of the most actively studied topics in the field, generally aims to drive a multi-agent system to achieve a desired geometrical pattern. Formation control has an impact on many real-world applications, like in autonomous Search and Rescue operations, environmental monitoring tasks, and ground and aerial coverage and surveillance. Several strategies to perform c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 432–440, 2021. https://doi.org/10.1007/978-3-030-58380-4_52

Stability and Reconfiguration for Multi-agent D-Formation Control

433

formation control can be found in the literature [3,8], but the approach that is usually employed is to a priori define a rigid target structure, and to try to enforce a set of strict constraints in order to bring each agent to its pre-assigned position. In this paper, it is presented a novel less-constrained approach, called Dformation control, that takes inspiration from the flocking of birds and exploits a virtual potential function to efficiently achieve specific formations. As it will be discussed, this approach removes the need to assign target positions beforehand, allowing each agent to naturally converge to a proper position. The objective of this work is to introduce, study and develop the D-formation control law. Specifically, the main contributions lie in the characterization of the equilibrium configurations of the D-formation control law and in the design of a generalized version of such controller, allowing for switching from one stable equilibrium to another one, that relies on the regularization paradigm.

2 2.1

Multi-agent Systems Modelling and Control Dynamics of a Fleet of Agents

An intuitive model for formation control is presented here. The context is that of a group of N agents moving in a two or three-dimensional Euclidean space. A common choice is to model each agent as a single integrator [6], i.e. q˙i = ui ,

qi ∈ Rf

(1)

where qi , i = 1, . . . , N , represents the agent’s position in 2D or 3D (respectively, f = 2, 3), and ui represents the agent’s input. These agents are said to be anonymous, in the sense that they all act alike and are identical. They are able to detect each other and, in particular, to measure their relative positions, but they have no knowledge about their absolute location. 2.2

Control of Flocks, Gradient-Descent of a Virtual Potential Function

As mentioned in the introduction, numerous strategies can be employed to control the behaviour of such group of agents. In this work, the focus is on a behavioural-based control approach based on gradient-descent. Gradient-descent control relies on two key ingredients. Firstly, a virtual potential function V (q) that represents the energy level of the whole N -agent flock in Rf is considered, where q ∈ RN f denotes the state of the system. Secondly, the assumption is made that each agent is modeled as a point mass with fully actuated dynamics, as assumed in Eq. (1). In this way, it is possible to obtain a so-called gradient system of the form: q˙ = −∇V (q)

(2)

434

A. Colotti et al.

Gradient systems, under shallow regularity conditions for V (q), are guaranteed to converge to an equilibrium [4], and they can greatly simplify the study of the critical configurations of the agents. The stability analysis is reduced to the critical points’ evaluation, and the subsequent spectrum analysis of the Hessian of a static potential function, as later discussed in Sect. 3. 2.3

Usual Geometry of Flocks: α-Lattices

In this section, we detail a usual strategy able to provide a formal objective for geometry when controlling flocks, called α-lattices [7]. For this strategy, a group of N mobile agents is considered, and it is sought the set of configurations q in which each agent i is equally spaced with respect to all other agents belonging to its neighborhood N (qi ). In terms of inter-agent distances, this geometric objective can be attained by the solutions of the following set of constraints: ∀j ∈ N (qi ). (3) qj − qi  = D, The solutions of the set of constraints in (3) play the role of desired conformations of agents in a flock. The configurations that respect these constraints are called α-lattices. In the following section, we detail our new strategy for the control of flocks, called the D-formation problem.

3 3.1

The D-Formation Problem Problem Formulation and Theoretical Analysis

Differently from the α-lattice formation problem, the D-formation problem can be stated as follows: Problem 1 ( D-formation). Find a distributed control law u(·) for a swarm of N anonymous agents described by model (1) such that each single agent aims to be at the same distance D from all the others. From a geometric point of view, the target formation, which will be called D-formation, can be defined as follows: qj − qi  = D,

∀j = i

(4)

It can be seen as a generalization of the α-lattice, where each agent’s neighbourhood contains the whole formation. However, with respect to the α-lattice, here it is assumed that the information about agents position can be spread throughout the network (e.g. via multi-hop procedures or message broadcast) and the rationale behind this choice is to prevent the creation of clusters of agents and

Stability and Reconfiguration for Multi-agent D-Formation Control

435

split formations, a risk that is inherently present in the objective stated by (3). Contrarily to α-lattices, however, the D-formation configuration can be exactly achieved for agents’ fleets up to N = 3 or N = 4 agents, considering the 2D or 3D case respectively. Therefore, in the remainder of the paper, D-formation configurations will indicate all equilibrium configurations that solve Problem 1 also in a soft sense (namely, not exactly). In order to try to reach the D-formation, a virtual potential function V (q) is defined with respect to a target inter-agent distance D as: ⎡ ⎤ N N 1 ⎣ 1  (5) (dij − D)2 ⎦ , V (q) = 2 i=1 N j=i

where dij = qj − qi  is the distance between agents i and j. The idea behind this virtual potential function is to embed the D-formation problem: having that V (q) is non strictly convex, its minima are D-formation configurations. In particular, the configurations corresponding to V (q) = 0 are related to the feasible exact ones, i.e. the solutions for which dij = D for any agents. Accordingly, the control law ui that makes the system a gradient system can be derived from (5), yielding: ui = −∇i V =

N 2  qj − qi (dij − D) . N dij

(6)

j=i

Clearly, although an exact D-configuration may not be physically achievable, thanks to the nature of the gradient control, the system will still converge to a point of local minimum, namely to a equilibrium configuration. Moreover, differently from the formation control approaches often found in the literature, where a target configuration is chosen and each agent is forced to go to its pre-assigned position, the D-formation control strategy simply relaxes an initial configuration to the local minima of the potential function V (q). Interestingly, the potential (5) can be decomposed into two terms as follows: ⎡ ⎡ ⎤ ⎤ N N N N     1 1 ⎣1 ⎣1 dij (dij − D)⎦ − D(dij − D)⎦, (7) V (q) = 2 i=1 N 2 i=1 N j=i j=i 

 

 V0 (q)

V+ (q)

where V0 (q) characterizes the equilibrium with a level of geometric symmetry (V0 (q) = 0)1 and V+ (q) allows to discern among the different equilibrium configurations. The analysis of the formations achievable by the proposed control law is performed, by exploiting the first order condition for the equilibria of the gradient 1

To attain V0 (q) = 0, the agent distances tend to balance among themselves between the set with dij > D and that with dij < D.

436

A. Colotti et al.

system. More formally, the critical points of V (q) can be found as the solutions of the unconstrained optimization problem: q ∗ = arg min V (q)

q ∗ : ∇V (q ∗ ) = 0

=⇒

q∈RN f

(8)

where {q ∗ ∈ RN f } represents the set of equilibrium configurations. 3.2

Numerical Analysis for Equilibria Evaluation

Given the complexity of the considered potential function, finding the equilibrium configurations q ∗ is performed using numerical tools. By employing the IBEX library [1], which is a C++ library for constraint processing over real numbers, the solutions of the system of equations given by (8) are sought and all the D-formation planar equilibria have been evaluated for fleets of up to 5 agents, and their stability has been assessed studying the spectrum of the Hessian of the potential function. Unfortunately, though, this approach is computationally costly and only small planar systems can be solved. Hence, in order to have a wider perspective on the controller’s capabilities, a second approach to evaluate local minima has been developed. This method is implemented as a Multi-Start procedure that exploits the trust-region algorithm [5], an iterative algorithm that evaluates a local minimum starting from a given initial configuration. The agents’ initial positions are picked uniformly over a 100 × 100 m surface in the planar case, or a 100 × 100 × 100 m cube in the 3D one. Obviously, we cannot guarantee that this approach is able to find all local minima, but it was able to identify several ones for the tested formations. These results are summarized with some examples in Table 1 and Table 2, showing also the values of the potential function. For all the simulations, a value D = 5 m has been chosen. Table 1. Examples of D-formation equilibria in 2D. Full set of equilibria for N = 3, 4; pair of stable equilibria for N = 10. N

V0 V+ V Stable

3

0 2.78 2.78 No

4

0 0 0 Yes

0 6.25 6.25 No

0 2.51 2.51 No

10

0 1.07 1.07 Yes

0 12.49 12.49 Yes

0 12.36 12.36 Yes

Stability and Reconfiguration for Multi-agent D-Formation Control

437

Table 2. Examples of D-formation equilibria in 3D, for N = 4, 5, 10. N

4

V0 V+ V Stable

4 4.1

0 6.25 6.25 No

0 1.07 1.07 No

5

0 0 0 Yes

0 2.64 2.64 No

10

0 0.82 0.82 Yes

0 12.49 12.49 No

0 5.74 5.74 Yes

D-Formation Switching Control Switching Formation Task

The control law (6) allows to move from a generic initial displacement of the agents towards one of the D-formation equilibria (local minima of V (q)), so as to solve Problem 1. It is now to understand whether it is possible to switch from one local minima to another one and converge to a specific chosen configuration, as usually done in formation control. Formally, the following problem can be stated as Problem 2 (Switching formation). Find a distributed control law u(·) for a swarm of N anonymous agents to switch between D-formations. whose solution objective is to switch between different stable configurations. To this aim, a regularized version of the potential (5) is employed, since it allows to favour particular agents’ arrangements exploiting a priori information on the equilibrium configurations. In practice, the idea is to add a regularization term R(q) to the potential function V (q), in order to penalize the initial unwanted equilibrium, which becomes a repulsive configuration. Thanks to R(q), the action of the controller forces the agents potential V (q) outside the basin of attraction of the initial equilibrium, to allow the convergence towards other configurations. 4.2

Fixed Target Controller

In this context, it is chosen to design a control law that drives the configuration towards a specific equilibrium q∗ , q∗ ∈ {q ∗ } that is assumed to be known. In order to achieve this task, the idea is to consider the following potential: Vc (q) = V (q) + R(q)

with

R(q) =

1 kc q∗ − q2 , 2

kc ∈ R+

(9)

Clearly, R(q) is a convex function with unique minimum represented by q∗ . Since a gradient-descent controller is employed, the only action of the regularization term is to push the formation towards the configuration q∗ , in particular when

438

A. Colotti et al.

the current configuration is distant from the desired q∗ . Being it an equilibrium configuration for V (q), q∗ is also an equilibrium for the whole potential Vc (q). The agents’ control law is then computed as follows: ⎡ ⎤ N 2 ⎣ qj − qi ⎦ + kc (qi∗ − qi ). (dij − D) (10) ui = −∇i Vc (q) = N dij j=i

At the beginning of the control action, when the formation is near q ∗ = q∗ , the effect of V (q) is negligible and each agent is driven by −∇i R(q) towards qi∗ . As the formation leaves the initial basin of attraction of the initial equilibrium, the fleet overcomes a relative maximum of V (q), to finally descend to the q∗ . It is worthwhile to note that an accurate tuning of the parameter kc is required, in order to guarantee the switching between the attraction basins of the two equilibria.

Fig. 1. Fixed target equilibrium controller. (a)–(d): 2D case with N = 5 and (e)–(h): 3D case with N = 4. Initial (a), (e) and final (b), (f) configurations; (c)–(g): evolution of the potential energy terms; (d)–(h): evolution of the coordinates of a single agent.

Figure 1 reports two examples of switching configuration task, related to a 2D case (switch between soft D-formations) and to a 3D case (switch towards an exact D-formation). The plots of the energy potentials (Fig. 1(c), (g)) allow to remark that local minima are present in V (q) (red curves) and the role of the regularization term R(q) in moving from the initial equilibria. Also, the final value of Vc (q), equal to that of V (q), is zero only in the case of exact D-formation (Fig. 1(g)).

Stability and Reconfiguration for Multi-agent D-Formation Control

5

439

Conclusions

In this work an approach to formation control has been proposed, based on the definition of a suitable potential energy function, which aims at solving the Dformation problem where all agents of a group aim at being at the same distance D from each other. This problem turns out to be a non-convex optimization problem, where the only exact solution reachable is the triangular configuration in 2D and the tetrahedron configuration in 3D, while multiple minima are solving the problem in a soft sense for generic number of agents in 2D–3D. The proposed solution to this problem exploits the theory of gradient systems and allows the design of a distributed control law that drives the multi-agent system to one equilibrium point within the set of available configurations; furthermore, this law can be adapted in order to reach a specific desired configuration through a switching equilibria control. However, from a practical point of view, it would be better to be able to privilege a set of equilibria, characterized by properties in common, more than a specific equilibrium configuration (chosen a priori ). This aspect, together with a strategy to automatically tune the gain of the switching controller is the subject of future study. Another possible line of research regards the in-depth study of the stability of the configurations: since the potential function is not convex, the presence of isolated minima with different values of V (q) has been shown but the analysis of these values alone is not sufficient to draw conclusions on the attraction basins, while this issue becomes important to optimize the switching formation control effort.

References 1. IBEX library website. http://www.ibex-lib.org/ 2. Alexis, K., Nikolakopoulos, G., Tzes, A., Dritsas, L.: Coordination of helicopter UAVs for aerial forest-fire surveillance. In: Valavanis, K.P. (ed.) Applications of Intelligent Control to Engineering Systems, pp. 169–193. Springer, Dordrecht (2009) 3. Anderson, B.D.O., Yu, C., Fidan, B., Hendrickx, J.M.: Rigid graph control architectures for autonomous formations. IEEE Control Syst. Mag. 28(6), 48–63 (2008) 4. Hirsch, M.W., Devaney, R.L., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra, vol. 60. Academic Press, San Diego (1974) 5. Mor´e, J., Sorensen, D.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 3, 553–572 (1983) 6. Oh, K.K., Park, M.C., Ahn, H.S.: A survey of multi-agent formation control. Automatica 53, 424–440 (2015) 7. Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control 51(3), 401–420 (2006). https://doi.org/10.1109/ TAC.2005.864190 8. Park, M.C., Ahn, H.S.: Distance-based acyclic minimally persistent formations with non-steepest descent control. Int. J. Control Autom. Syst. 14(1), 163–173 (2016)

440

A. Colotti et al.

9. Prokaj, J., Medioni, G.: Persistent tracking for wide area aerial surveillance. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2014 10. Shamma, J.: Cooperative Control of Distributed Multi-agent Systems. Wiley, Hoboken (2008) 11. Yang, H., Han, Q., Ge, X., Ding, L., Xu, Y., Jiang, B., Zhou, D.: Fault-tolerant cooperative control of multiagent systems: a survey of trends and methodologies. IEEE Trans. Ind. Inform. 16(1), 4–17 (2020) 12. Zhao, S., Dimarogonas, D.V., Sun, Z., Bauso, D.: A general approach to coordination control of mobile agents with motion constraints. IEEE Trans. Autom. Control 63(5), 1509–1516 (2018)

Braking of a Solid Body Supported by Two Supports on a Horizontal Rough Plane Marat Dosaev1(B) , Vitaly Samsonov1 , Liubov Klimina1 , Boris Lokshin1 , Shyh-Shin Hwang2 , and Yury Selyutskiy1 1 Institute of Mechanics of the Lomonosov, Moscow State University, Moscow, Russia

[email protected] 2 Chien Hsin University of Science and Technology, Taoyuan, Taiwan

Abstract. Dry friction between elements of mechanisms remains an important aspect of theoretical and applied problems of machine science. In many cases of mathematical modeling, to overcome the paradoxes associated with the application of the Coulomb law in describing the friction between the elements of mechanisms, it is necessary to take into account the flexibility of these elements. The problem of sliding a heavy body with two elastic support legs along a horizontal rough plane is considered. The support legs are massless rods that resting on the plane at one end. At the other end, the rods are connected to the body by means of viscoelastic springs. It is shown that with absolutely rigid fastening of the leg guides, it is impossible to balance the body when any type of asymmetry occurs. Sliding friction is described using the Coulomb law. Several different types of motion are found: sliding the legs in one direction, sliding the legs in different directions, sliding of one leg (rotation around a fixed point). Keywords: Mathematical modelling · Friction paradoxes · Coulomb law · Elastic support · Nonlinear dynamic system

1 Introduction When modeling mechanical systems, it is necessary to take into account dry friction between the elements of the mechanisms. Items of the applicability of the Coulomb law were seriously questioned at the end of the 19th century – beginning of 20th century in a scientific discussion [1–5], which raised problems that became known as the “Painlevé paradoxes.” One of the paradoxes [6, 7] is that slipping can stop due to a tangential impact, or in other words due to a friction shock. Similar problems are described, for example, in the problem of the contact of the brake shoe with the wheel [8] and confirmed by experiments [9]. Similar paradox was described, for example, in the problem of the contact of a brake shoe with a wheel [10–12]. However, given the flexibility of the contacting bodies, it is possible to overcome these problems. In particular, in the brake shoe problem, the contact was modeled using an elastic spring. In this problem [12], interesting effects were found in the interaction of the elastic force and the friction force. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 441–448, 2021. https://doi.org/10.1007/978-3-030-58380-4_53

442

M. Dosaev et al.

It is of certain interest to study in more detail the processes that occur directly at the points of contact of the body with a rough support. The sliding of a body along a horizontal plane was considered by many authors (for example, [13, 14]). Some authors consider the motion of robots resting on a rough surface by several legs. There are various types and prototypes of legged robots in the literature, namely: bipeds, tripods, quadrupeds etc. (e.g. [15–17]). In recent years, new types of robotic devices that are driven by internal masses and internal forces have attracted the attention of scientists. There are many possible applications for this type of autonomous robots. In [18], the design of such an inertial rectangular robot with one unbalanced rotor and one flywheel was proposed. The body of the robot slides along a rough plane. If the robot body is in contact with the supporting surface along its entire bottom part, then the point of application of the normal reaction is unknown. Therefore, to regularize the description of the behavior of such a robot, one of the possible options for the design of the robot can consider the support of the robot on two elastic legs. In the task of developing a control algorithm for such an elastic system with dry friction, the following problem will arise: to compensate for the coupling reaction in the hinges, an infinite value of the tangential reaction at the contact points is necessary. Dry rest friction is not able to keep at least one of the legs from slipping. In this paper, the problem of sliding a heavy body with two elastic supports along the horizontal plane is considered. Sliding friction is described using the Coulomb law. Some paradoxes of friction are discussed including new type of features of elastic systems with dry friction. It is shown that during sliding several different types of motion are possible.

2 Friction Paradoxes We consider a heavy body ABCD (Fig. 1), with which guide bearings are rigidly connected. P is the weight, AB = 2a, AD = 2b. On guides without friction, inflexible massless rods (legs) glide at one end resting on a horizontal rough plane.

a

b

c

Fig. 1. The body with fixed support points on rough plane.

At the other end, the rods are connected to the body by means of viscoelastic springs (k is the spring’s stiffness). If the ends of the legs are fixed (A1 B1 = AB), then the body can only perform translational movement vertically. The equality of the stiffness of the springs in this case ensures the equality of the vertical reactions of the

Braking of a Solid Body Supported by Two Supports on a Horizontal Rough Plane

443

supports: N 1 = N 2 = P/2. If the body is homogeneous, then the system has an axis of dynamic inertia. So, there is an obvious equilibrium position. However, if the center of mass G of the body is shifted horizontally (Fig. 1a), it turns out that the equilibrium conditions are not satisfied! Reactions do not compensate for the moment around the center of mass. An even more interesting case is if a small horizontal force Fe is applied to a symmetrical body (Fig. 1b). In this case, at the points A1 , B1 of support, horizontal reactions Ffr1 , Ffr2 occur, directed in the direction opposite to the direction of the lateral force Fe . Once again, there is nothing to compensate for the moment of these reaction forces relative to the center of mass and, again, equilibrium is impossible. Moreover, it is impossible to realize these reactions by dry friction! This is where the peculiarity of the “paradox” type is manifested.

3 Influence of Leg Stiffness To regularize the system, we add a degree of freedom. We add a cylindrical hinge to the fastening of one of the legs. We equip this hinge with an elastic spiral (coil) spring (Fig. 1c) to achieve the necessary stiffness c. Supporting rods can deviate from the vertical at different angles ϕ1 and ϕ2 , the value of the torque T2 of the spiral spring is T2 = c(ϕ2 − ϕ1 ). We introduce two general coordinates: ϕ = ϕ1 , x = l0 − l, here l0 is the length of the rods with unperturbed springs. Active forces acting on the body are potential,  and  the equilibrium  position is determined from the following system of equations: ∂ ∂ϕ = 0, ∂ ∂x = 0 , where  = P + T2 + el1 + el2 + F e is the potential energy of the system,  = mgyG (ϕ, x) + c(ϕ2 (ϕ, x) − ϕ)2 /2 + kx2 /2+ k(l2 (ϕ, x) − l0 )2 /2 − F e xG (ϕ, x), xG , yG are the coordinates of the center G of mass, l2 = BB1 . The dependence of the equilibrium position on the coordinates is a complex function of coordinates. We consider the stiffness c of the coil spring to be a sufficiently large value, we expand the solution of equilibrium equations in a series of c−1/3 : ϕ = a1 c−1/3 + a2 c−2/3 + a3 c−1 + o(c−1 ), x = mg/2k + b1 c−1/3 + b2 c−2/3 + O(c−1 ), where the coefficients a1 , a2 , a3 , b1 , b2 depend on the parameters of the problem. We fr find for this approximation the values of horizontal reactions: F2 = aϕ 2 c/((2aϕ + fr fr x)l0 ) + O(ϕ), F1 = F e − F2 . It can be seen that with c tending to infinity, horizontal reactions tend to infinity. The necessity of applying an infinite horizontal reaction for equilibrium in the case of an absolutely rigid spiral spring determines the impossibility of realizing this reaction by dry friction. Any reason that causes asymmetries in the problem statement will cause at least one of the legs to slip.

4 Body Sliding on a Support In the following, we consider the mechanical system without a coil spring. Both legs can slide over the surface of the support. The system generally has three degrees of freedom. But at times, when one of the legs stops, additional constraints arise that reduce the number of degrees of freedom by one. Let the lateral force F e ≡ 0. Initially the mechanical system is in equilibrium.

444

M. Dosaev et al.

We introduce a new coordinate system (see Fig. 2). The axis Ox is parallel to the plane of the support and is located at a distance l0 + b from it. The Oy axis passes through the center of mass G in the equilibrium position of the system. In the initial position, the coordinates of the center of mass G of the system are as follows: (xG , yG ) = (0, −mg/2k). We impact some initial horizontal velocity VxG 0 to the center G of mass of the system. For definiteness, we assume that the initial velocity is directed along the vector A1 B1 : VA1 = x˙ A1 > 0, VB1 = x˙ B1 > 0. Hereafter, we denote the time derivative by the dot. Thus, we will call the rod BB1 (|BB1 | = l2 ) the front leg, and the rod AA1 (|AA1 | = l1 ) the back leg. Due to the presence of friction, the body must stop for some time. Such braking process can be divided into several phases of motion: sliding on two legs; instantly stopping one leg and then sliding on two legs in different directions; and stopping one leg and sliding on the other.

5 Sliding of Two Legs Let both legs slide on the support (Fig. 2). We assume that the tangential reactions of the support at the sliding points depend on the normal reactions according to the Coulomb fr fr law: F1 = μsgn(VA1 )|N1 |, F2 = μsgn(VB1 )|N2 |, μ is the friction coefficient. As the generalized coordinates, we choose the coordinates of the center of mass xG , yG and the angle ϕ of the body inclination. We write the equations of motion from the theorem on the motion of the center of mass of the mechanical system and the angular momentum theorem relative to the center of mass. fr

fr

m¨xG = −F1 − F2 m¨yG = −mg + N1 + N2 J ϕ¨ = N1 ((l1 + b) sin ϕ + a cos ϕ) − N2 (a cos ϕ − (l2 + b) sin ϕ) + fr

fr

(F1 + F2 )(l0 + b + yG )

(1)

Here J is the central moment of inertia of the body. We determine the values of the normal reactions at the support points as follows: N1 = −Fel1 cos ϕ, N2 = −Fel2 cos ϕ, where values of the forces Fel1 , Fel2 , with which springs act on the body, are the following: Fel1 = −k(l1 − l0 ) − h˙l1 , Fel2 = −k(l2 − l0 ) − h˙l2 , h is the viscosity coefficient of the springs. Note that the right-hand side of the second equation of system (1) does not depend on the coordinate xG and velocity x˙ G . The system (1) was integrated numerically. For calculation, as an example, hereinafter the following set of parameters was taken: b = 100, a = 200, l0 = 5, m = 0.1, k = 1, g = 10, J = 1, μ = 0.44, h = 0.0001. The example of calculated time dependences of the problem variables during two legs sliding are shown in Fig. 3 for the initial horizontal velocity VxG 0 = 1.

Braking of a Solid Body Supported by Two Supports on a Horizontal Rough Plane

445

Fig. 2. The body gliding on rough plane.

6 Instant Stop of One Leg Let at some point in time the speed of the reference point A1 of the back leg A1 A becomes equal to zero. This means that there is an instant stop of this point. It is necessary to determine whether this point will continue to stop (due to the fact that the horizontal reaction is within the angle of friction) or whether it will begin to move in the opposite direction. Let us write out the dependence of the coordinates of the center of mass G on the position of point A1 and the angle of inclination: xG = xA1 + (l1 + b) sin ϕ + a cos ϕ;

yG = −l0 − b + (l1 + b) cos ϕ − a sin ϕ

Hence, for horizontal acceleration of the center of mass, we have: x¨ G = x¨ A1 − ¨ 1 + b) cos ϕ − a sin ϕ) + ˙l1 cos ϕ ϕ˙ + ¨l1 sin ϕ. If ϕ˙ 2 ((l1 + b) sin ϕ + a cos ϕ) + ϕ((l the leg AA1 fell into the angle of friction, then at the stopping time and subsequent time instants xA1 ≡ const ⇒ x˙ A1 ≡ x¨ A1 ≡ 0. From here, the horizontal acceleration of the center of mass, when AA1 continues to stop, can be as follows: x¨ Gstop_A1 = −ϕ˙ 2 ((l1 + b) sin ϕ + a cos ϕ) + ϕ((l ¨ 1 + b) cos ϕ − a sin ϕ) + ˙l1 cos ϕ ϕ˙ + ¨l1 sin ϕ (2) Following the first equation of (1) we determine the horizontal reaction at the back   fr fr point A1 : F1 = m¨xGstop_A1 − F2 . Calculate the following difference: 1 = μN1 − Ffr1 . The back leg will rest only if the following inequality is held: 1 > 0. If this inequality is not fulfilled, then after an instant stop, the leg will begin to slide in the opposite direction. Numerical calculations show that 1 < 0 at the time moment when the speed of point A1 becomes equal to zero. This means that the speed of point A1 changes its sign, and this point begins to move in the opposite direction.

7 Gliding of One Leg (Motion Around a Fixed Point) After the back leg stopped instantly, the front leg BB1 also stopped. We write the dependence of the coordinates of the center of mass on the position of point B1 and the angle

446

M. Dosaev et al.

of inclination ϕ: xG = xB1 + (l2 + b) sin ϕ − a cos ϕ., Differentiating this equality and taking into account that x¨ B1 = 0, similarly to (2), we get the horizontal acceleration of the center of mass when the point B1 stays at rest: ¨ 2 + b) cos ϕ + a sin ϕ) + x¨ GstopB1 = − ϕ˙ 2 ((l2 + b) sin ϕ − a cos ϕ) + ϕ((l ˙l2 cos ϕ ϕ˙ + ¨l2 sin ϕ (3) Following the first equation of (1) we determine the horizontal reaction at thefront   fr  fr fr point B1 : F2 = m¨xGstopB1 − F1 . Calculate the following difference: 2 = μN2 − F2 . Calculations show that when the front leg stops (VB1 = 0), 2 > 0. Thus, the front leg enters inside the friction angle and will rest at subsequent instants of time. We fix the values of the generalized coordinates at the moment of stopping point B1 : xG = xG∗ ,yG = yG∗ , ϕ = ϕ ∗ . In this case, the horizontal coordinate of point B1 : xB1 = xB∗ 1 = xG∗ −(l2 +b) sin ϕ ∗ +a cos ϕ ∗ = const. For further calculation, it is necessary to switch to motion around a fixed point. Reactions do not compensate for the moment around the center of mass. The mechanical system has lost one degree of freedom, and, accordingly, two generalized coordinates are sufficient to describe its state. Choose two coordinates: yG , ϕ. We use the following dynamic system that describes the motion of the body: m¨yG = −mg + N1 + N2 J ϕ¨ = N1 ((l1 + b) sin ϕ + a cos ϕ) − N2 (−(l2 + b) sin ϕ + a cos ϕ) − m¨xGstopB1 (l0 + b + yG )

(4) In addition to the system (4), the horizontal coordinate and velocity of the center of mass can be obtained using the following kinematic relations: xG = xB∗ 1 + (l2 + b) sin ϕ − a cos ϕ, x˙ G = ϕ((l ˙ 2 + b) cos ϕ + a sin ϕ) + ˙l2 sin ϕ.

Fig. 3. Time dependences during body sliding on both legs of the horizontal coordinate of the center of mass xG (a), angular speed ϕ˙ (b), and the velocity VB1 of the point B1 (c).

8 Discussion The stop of both legs is possible only at zero angular velocity of the body. The braking of the body on elastic supports in the absence of a driving lateral force leads to oscillations of the center of mass relative to a certain position: (xGfin , yGfin ).

Braking of a Solid Body Supported by Two Supports on a Horizontal Rough Plane

447

For comparison, we consider the braking of a flat body (a body on rigid supports) along a rough horizontal plane with dry friction. Let us estimate the braking time 2 /2 + V0 tbraking , by which tbraking = V0 /(μg) and the distance xbraking = −μgtbraking the body will shift during this braking at a given initial speed V0 . It turns out that the horizontal displacement xGfin of the center of mass of the body on two elastic supports as a result of such braking is less than that during braking of the body on rigid supports. Moreover, the braking time of the body on two supports is significantly longer than the braking time of the body on hard supports.

9 Conclusions The problem of sliding a heavy body with two elastic support legs along a horizontal rough plane is considered. It is shown that with absolutely rigid fastening of the leg guides, it is impossible to balance the body when any type of asymmetry occurs. Sliding friction is described using the Coulomb law. During sliding, several different types of movements are possible: sliding the legs in one direction, sliding the legs in different directions, sliding of one leg (rotation around a fixed point). Comparison of motion behavior of the system in consideration with body sliding without support is given. Acknowledgments. This work was partially supported by the Russian Foundation for Basic Research, projects N 18-01-00538.

References 1. Painlevé, P.: Leçon sur le frottement. Hermann (1895) 2. Sparre, De: Sur les lois du frottenent de glissement. Comptes Rendus 141, 310 (1905) 3. Klein, F.: Zur painleves kritik der coulombschen reibungsgesetze. Ztschr. f. Math. u. Physik 58, 186 (1909) 4. Zur, Mises R., der Reibungsgesetze, Kritik: Ztschr. f. Math. u. Physik 58, 191 (1909) 5. Hamel, G.: Bemerkungen zu den vorstehenden Aufsatzen der Herren F. Klein und R. v. Mises. Ztschr. f. Math. u. Physik, 58, 195 (1909) 6. Lecornu, L.: Sur le frottement de glissement. Comptes Rendu des Seances de l’Academie des Sciences, 140(6), 635–637, 847–848 (1905) 7. Bolotov, Y.A.: The motion of a plane material under constraints with friction. Mat. Sbornik 25(4), 562–708 (1906) 8. Samsonov, V.A.: Essays on the Mechanics. Nauka, Moscow (1980) 9. Ivanova, T.B., Erdakova, N.N., Karavaev, Y.L.: Experimental investigation of the dynamics of a brake shoe. Doklady Phys. 61(12), 611–614 (2016). https://doi.org/10.1134/S10283358 16120028 10. Neimark, Y.I., Fufayev, N.A.: Painlevé paradoxes and the dynamics of a brake shoe. Prikl. Mat. Mekh. 59(3), 366–375 (1995) 11. Samsonov, V.A.: The dynamics of a brake shoe and “impact generated by friction”. Prikl. Mat. Mekh., 69(6), 912–921 (2005) 12. Dosaev, M.: On instability of a brake shoe. In: Proceedings of the 14th IFToMM World Congress (2015). https://doi.org/10.6567/iftomm.14th.wc.os4.015 13. Macmillan, W.D.: Dynamics of Rigid Bodies. McGraw-Hill book co, Inc, NY (1936)

448

M. Dosaev et al.

14. Samsonov, V.A.: To the 100th anniversary of the result E.A. Bolotova. J. Mach. Manuf. Reliabil. 2, 100–102 (2006) 15. Rozenblat, G.M.: Motion of a body sliding on a rough horizontal plane and supported at three points. Doklady Phys. 62(9), 430–433 (2017). https://doi.org/10.1134/S1028335817090038 16. Agheli, M., Qu, L., Nestinger, S.S.: SHeRo: scalable hexapod robot for maintenance, repair, and operations. Robot Cim-Int. Manuf. 30, 478–488 (2014) 17. Grzelczyk, D., Sta´nczyk, B., Awrejcewicz, J.: Prototype, control system architecture and controlling of the hexapod legs with nonlinear stick-slip vibrations. Mechatronics 37, 63–78 (2016). https://doi.org/10.1016/j.mechatronics.2016.01.003 18. Dosaev, M., Samsonov V., Hwang Sh.-Sh.: Friction coefficient estimating in problem of planar motion of a friction powered robot. In Proceedings of 15th Conference on Dynamical Systems: Theory and Applications (DSTA 2019), Theoretical Approaches in Non-Linear Dynamical Systems 1, pp. 125–132 (2019)

Workspace Analysis and Torque Optimization on a Sch¨ onflies-Motion Generator Bruno Belzile(B)

and Jorge Angeles

Department of Mechanical Engineering and Center for Intelligent Machines, McGill University, 817 rue Sherbrooke Ouest, room 270, Montreal, QC H3A 0C3, Canada {bruno,angeles}@cim.mcgill.ca

Abstract. Pick-and-place robots are used to complete many tasks in industry, from packaging operations to manufacturing. Sch¨ onflies-motion generators (SMG), capable of three translations and one rotation about an axis of fixed direction, are commonly used for this purpose. Many researchers have focused in recent years on finding new mechanisms to reduce the time necessary to complete a pick-and-place operation, as well as providing a higher rotability of the mobile platform, an issue that handicaps several SMGs. In this paper, the workspace of a parallel isostatic SMG, with virtually unlimited rotability, is analyzed in order to optimize a paradigm trajectory with the objective of reducing the torque needed from the motors, thus allowing for higher operation frequencies. The paradigm path is kept unchanged, but the starting-point location is optimized. It is shown this location significantly affects the maximum absolute torque required to cross the same distance with the same cycle time.

Keywords: Sch¨ onflies-motion generator Torque

1

· Kinematics · Dynamics ·

Introduction

Sch¨ onflies-motion generators (SMG) are capable of four degree-of-freedom operations, i.e., three independent translations and one rotation about an axis of fixed direction [12]. These are often called SCARA systems, in reference to the functionality of the well-known Selective-Compliance Assembly Robot Arm, capable of such motion with one prismatic and three revolute joints in a serial array [13]. SCARA systems are commonly used to perform pick-and-place operations (PPO), common in packaging tasks. The fastest commercial SMGs are generally parallel-kinematics machines or parallel-serial mechanisms, including the threelimb Delta architecture with a telescopic Cardan shaft [4] as well as the four-limb Supported by NSERC (Canada’s Natural Sciences and Engineering Research Council) through the Postdoctoral Fellowship Program and grant No. 4532-2010. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 449–457, 2021. https://doi.org/10.1007/978-3-030-58380-4_54

450

B. Belzile and J. Angeles

H4, I4L, I4R, Heli4 and Par4 [16]. The Adept Quattro s650H is a realization of the Par4 architecture, capable of three cycles per second. On the one hand, an extensive literature on trajectory-planning with robotic manipulators [5,8] is available. A large range of performance indices has been proposed to meet different criteria. Several control algorithms, including to avoid torque saturation, have also been reported [18]. On the other hand, some pickand-place robots have also been designed and optimized with respect to a particular trajectory [7]. However, while extensive studies on kinematics (forward and inverse kinematics, workspace and singularity analyses) can be found, as well as some papers on their dynamics, trajectory-planning for SMGs with the objective of reaching higher velocities has received less attention. This work focuses on one particular parallel isostatic SMG, the Peppermill Carrier (PMC), which features only two limbs [3]. This robot is based on the CRRHHRRC1 closed kinematic chain [11]. It was designed to overcome one common drawback of parallel SMGs, i.e., the limited rotability of their moving platform (MP). Some architectures have been proposed by researchers to achieve the same objective. Arian et al. proposed an overconstrained four-limb manipulator capable of infinite rotability [1]. Wu et al. reported, in turn, a four-limb SMG capable of full circle end-effector rotation [19]. Schreiber and Gosselin used the kinematic redundancy of their design to generate unlimited rotation capabilities at the MP [17]. As mentioned above, the PMC stands out with isostaticity and its reduced number of limbs. In previous papers, the velocity scheduling of the PMC and other robots has also been optimized for PPOs [2,10,15]. With the same objective of minimizing the risk of torque saturation, this paper focuses on the workspace analysis of the PMC. This is done to find the optimum subworkspace to conduct a pick-and-place trajectory, as well as the optimum starting point. First, a description of the PMC, along with its kinematics and dynamics, is recalled for the sake of completeness. The test trajectory is then introduced and the PMC workspace is analyzed. Simulation results based on a dynamics model are reported. A subdivision of the workspace where the object to be picked-and-placed should be optimally located is also determined.

2

Kinematics and Dynamics

The kinematics and dynamics of the PMC, recalled here for completeness, were reported in a previous paper [9]. The architecture is depicted in Fig. 1. It consists of two strain-wave-gear-enhanced cylindrical drives (CSWG -drives), two limbs and the peppermill, to which a MP is attached. The CSWG -drives are differential mechanisms capable of generating translational and rotational motion of the cylindrical type independently, with the former along a direction parallel to the axis of the latter. The vector d is an array defining the translation and the rotation produced by the CSWG -drives, represented, respectively, by ui and θi , 1

R, H and C stand for revolute, helical (screw) and cylindrical joint, correspondingly, underlined characters denoting actuated joints.

Workspace Analysis and Torque Optimization on a SMG

451

Fig. 1. Schematic view of the PMC

where i = 1, 2. The 4 × 4 Jacobian matrix JC maps d into the array of joint variables ψ: (1a) ψ = JC d with ⎡

 ψ ≡ ψ1L ψ1R ψ2L

2π T 1⎢ −2π ψ2R , JC ≡ ⎢ p⎣ 0 0

pG 0 pG 0 0 2π 0 −2π

⎤ ⎡ ⎤ 0 u1 ⎢ θ1 ⎥ 0 ⎥ ⎥, d ≡ ⎢ ⎥ ⎣u2 ⎦ p G⎦ θ2 pG

(1b)

The indices 1L, 1R, 2L, and 2R stand, respectively, for the left- and righthand screws of the first and the second CSWG -drives. The four-dimensional vector T  x ≡ xC , yC , zC , φ defines the pose of the MP. The nonlinear equations to obtain d from x can be found in an earlier paper [9]. The pitch of the right- and the left-hand screws of the CSWG -drives are, respectively, p and −p, while the pitch of the right- and the left-hand screws of the peppermill are, respectively, q and −q. The gear-reduction ratio of the C-drives is G. The PMC is made up of 11 rigid bodies: four screws, two arms, two forearms, two nuts, and the peppermill. The torque vector τ is computed from the dynamics model: ¨ + Cψ˙ − γ − η τ = Iψ

(2a)

with 

τ ≡ τ1L τ1R τ2L τ2R

T

, I≡

11

Ti T Mi Ti ,

(2b)

i=1

C≡

11

i=1

˙ i + Ti T Wi Mi Ti ), γ + η ≡ (Ti T Mi T

11

i=1

E Ti T (wG i + wi ),

(2c)

452

B. Belzile and J. Angeles



 ∂ti Ii O Ωi O Mi ≡ , Wi ≡ , Ti ≡ O mi 1 O O ∂ ψ˙

(2d)

where 1 and O denote the 3 × 3 identity and zero matrices. The angular-velocity matrix Ωi is used to compute the angular-velocity dyad Wi of the i th body, as per Eqs. (2d). The former is the cross-product matrix of the angular-velocity vector ωi . The twist-shaping matrix of the i th body, Ti , maps the vector array ˙ into the twist vector ti of the i th body. Matrices Ii and of the motor speeds, ψ, Mi are, respectively, the inertia tensor and the inertia dyad [14] at the center E of mass (c.o.m.) of the i th body. Finally, wG i and wi are the gravitational and external wrenches exerted on the i th body, γ and η being their corresponding generalized-force vectors. The twist ti of and the wrench wi exerted on the i th body are evaluated at the body c.o.m. Vectors ni and fi are the moment and the force acting on the i th body, the former about, the latter at the body c.o.m.

3

Workspace Analysis

The trajectory used in this paper is a smoothed version of the Adept test cycle [6], i.e. an upward translation of 25 mm, followed by a horizontal 300-mm translation with a concurrent 180◦ rotation about the MP vertical axis, then a downward 25-mm translation. The complete cycle consists of the foregoing program and the symmetric motion back to the starting point P0 (x0 , y0 , z0 )2 . Before conducting the workspace analysis, the optimization criterion must be defined. To avoid motor saturation, the maximum absolute torque τmax is to be minimized, i.e., (3) global min (τmax (P0 , α)) P0 ∈W 0≤α≤π/2

the above expression being a function of the starting point P0 within the workspace W and the angle α the trajectory plane makes with the X-axis. Both variables are depicted in Fig. 2. For each pair (P0 , α), the maximum absolute value of the torque is computed from the expression τmax (P0 , α) = max (f (t)), 0≤t≤T

f (t) = max{|τ1L |, |τ1R |, |τ2L |, |τ2R |}

(4)

The first issue to consider is how changing the starting point location P0 changes the maximum absolute torque τmax required to complete a trajectory in a given time. On the one hand, if we consider only the translation of the C-drive alone, the distance to travel is a function of the angle α. In order to reduce the torque required, the former must be minimized. Considering the symmetry of the PMC architecture, this can be done by having a trajectory within a plane making an angle α = π/4. On the other hand, since the optimum posture of one limb will necessarily be the same for the other one, the optimum trajectory plane must thus include the origin O of the reference frame shown in Fig. 1. 2

The path and the scheduling, i.e. the fraction of the cycle time required to reach every intermediate point along the trajectory, remain unchanged in the examples.

Workspace Analysis and Torque Optimization on a SMG

453

Fig. 2. Trajectory plane within the workspace of the PMC: (a) 3D view with the Adept test cycle in red; (b) view from above

This statement is valid for any trajectory (not only the Adept test cycle) that is contained in a single vertical plane, which is usually the case if there is no obstacle in the workspace. However, in an industrial environment, objects to be picked up are not necessarily placed in one single location of one’s choice. The one parameter that the designer can control, though, is location of the horizontal plane where pick-andplace operations take place. Therefore, the height of this plane relative to the origin O is optimized to minimize the torque peaks. To this end, the workspace of the PMC is evenly discretized along the X−, Y − and Z−axes. With the model introduced in Sect. 2, for each point within the workspace, the Adept test cycle is completed for a set A of evenly distributed angles {αi } and the mean value of τmax is computed for that point, excluding trajectories that go through a singularity. In this light, the performance index for each selected point (x0 , y0 , z0 ) is defined as I(x0 , y0 , z0 ) ≡

n

|τmax (P0 , αi )| i=1

n

,

αi =

2π (i − 1), n

n>1

(5)

The results are depicted in Fig. 3(a) for an operation frequency3 of 1 Hz and n = 128. The data are illustrated for different horizontal places. It should be noted that if a trajectory is not feasible for a particular αi , the resulting τmax is disregarded and the value of n is lowered accordingly. The subworkspace above the XOY -plane is not considered, since singularities of the second kind are found above z = 0, thus reducing the effective workspace. The mean and maximum values as well as the standard deviation of I(x0 , y0 , z0 ) as functions 3

The operation frequency is the number of cycles per second conducted along one particular trajectory.

454

B. Belzile and J. Angeles

of z (defining the plane where the pick-and-place operations are conducted) are shown in Fig. 3(b). The size of the reachable area within these horizontal planes is also depicted in the same figure. 0.9

3.5 3

-0.05

-0.2

[Nm]

[Nm]

-0.15

max

z [m]

2.5

0.85

-0.1

0.34 0.32

2

0.3

1.5

0.28

1

0.26

0.5

0.24

W [m2]

0

0.36 mean(I) max(I) std(I) W

0.8

-0.25 -0.3 -0.35 0.5

0.4 0.4

0.3

0.2

y [m]

0.2 0.1

0

a

0

x [m]

0.75

0 -0.4

0.22 -0.3

-0.2

-0.1

0

z [m]

(b)

Fig. 3. (a) I(x0 , y0 , z0 ) for different values of z0 within the workspace of the PMC (1 Hz); (b) Mean and maximum values as well as standard deviation of I(x0 , y0 , z0 ) over horizontal planes as functions of the position of the latter (z) as well as the size of the reachable area (W )

It can be inferred from these results that the mean value of the performance index I slightly increases when z approaches 0. However, the maximum value significantly increases. This means that certain trajectories with starting points close to z = 0 will result in problematic torque peaks. Similarly, the maximum absolute torque for a trajectory conducted at operation frequencies of 1 Hz and 3 Hz can also be computed for the optimum vertical plane mentioned above (α = π/4, crossing the X-axis at the origin). The results are illustrated as functions of the starting point in Fig. 4. Apparently, while there is an absolute minimum (indicated by the green cross), there is also a set of points where τmax is close to the minimum. These points define a curve visible in Fig. 4 for both operation frequencies. By looking closer at the architecture of the PMC and its posture, one can infer that one particular posture is close to optimum. Indeed, all the points on this optimum curve result in nearly the same λi (absolute angle of the forearm, illustrated in Fig. 1). Isocontours for different angles λ1 are shown in the same figure. It should be noted that, as can be seen in Fig. 4 and mentioned in an earlier paper [2], τmax increases by the square of the operation frequency.

Workspace Analysis and Torque Optimization on a SMG

455

0.8 0.78

5.2 0 -0.05

0.05

5.2

0.76

0

5 0.74

4.8

-0.05

0.7 0.68

-0.15

0.66

-0.2

z [m]

4.6

6.6

5 4.8

6.4 [Nm]

4.4

max

z [m]

0.72

-0.14.2

6.8

5.4

-0.14.2

4.4

4.6

-0.15 -0.2

[Nm]

5.4

6.2

max

0.05

6

0.64

-0.25

-0.25 0.62

0.25

0.3

(x2+y2) [m]

(a) 1 Hz

-0.35 0.15

4.2

4.2 0.2

0.6

5.8

-0.3

4.4

-0.35 0.15

4.4

-0.3

5.6

0.2

0.25

0.3

(x2+y2) [m]

(b) 3 Hz

Fig. 4. τmax for α = π/4; the red cross denotes the location of the original starting point; the green cross is the location of the optimum; the isocontours are displayed over for λ1 [rad]

4

Discussion and Conclusions

We analyzed the workspace of a parallel isostatic SMG to find its optimum subdivision to conduct one specific trajectory, the Adept test cycle. The latter was chosen because it is the industry standard used by the robotics community to evaluate pick-and-place robots. As the above results show, the subworkspace located under the (z ≈ −0.1) plane is more appropriate for fast PPOs. Indeed, the maximum absolute torque required to conduct the Adept test cycle is generally lower than in the subworkspace located above. Moreover, a singularity of the first kind can be encountered. However, the reachable area decreases by the square of z. This methodology can be applied to any trajectory, Moreover, alternative parameter, such as the RMS value of the actuation torques, could have been used. The above results also confirm that the α = π/4 plane is the optimum. It was shown as well that the starting point of the trajectory significantly affects the maximum absolute torque required from the actuators. This is done to avoid saturation, reduce vibration and improve trajectory-tracking.

456

B. Belzile and J. Angeles

References 1. Arian, A., Isaksson, M., Gosselin, C.: Kinematic and dynamic analysis of a novel parallel kinematic Sch¨ onflies motion generator. Mech. Mach. Theory 147 (2020). https://doi.org/10.1016/j.mechmachtheory.2019.103629 2. Belzile, B., Angeles, J.: Heuristic algorithm for velocity scheduling with a Sch¨ onflies-motion generator. In: IFToMM World Congress on Mechanism and Machine Science, pp. 2411–2419. Springer, Krakow, Poland (2019). https://doi. org/10.1007/978-3-030-20131-9 238 3. Belzile, B., Karimi Eskandary, P., Angeles, J.: Workspace determination and feedback control of a pick-and-place parallel robot: analysis and experiments. IEEE Robot. Autom. Lett. 5(1), 40–47 (2020). https://doi.org/10.1109/LRA.2019. 2945468 4. Clavel, R.: Device for the movement and positioning of an element in space (1989) 5. Dai, Z., Sheng, X., Hu, J., Wang, H., Zhang, D.: Design and implementation of B´ezier curve trajectory planning in DELTA parallel robots. In: Liu, H., Kubota, N., Zhu, X., Dillmann, R., Zhou, D. (eds.) Intelligent Robotics and Applications. ICIRA 2015. vol. 9245, pp. 420–430. Springer, Cham (2015). https://doi.org/10. 1007/978-3-319-22876-1 36 6. Gauthier, J.F., Angeles, J., Nokleby, S.: Optimization of a test trajectory for SCARA systems. In: Advances in Robot Kinematics: Analysis and Design, pp. 225–234. Springer, Cham (2008) 7. Germain, C., Caro, S., Briot, S., Wenger, P.: Optimal design of the IRSBot-2 based on an optimized test trajectory. In: Proceedings of the ASME Design Engineering Technical Conferences 6A (2013). https://doi.org/10.1115/DETC2013-13037 8. Gosselin, C., Foucault, S.: Dynamic point-to-point trajectory planning of a twoDOF cable-suspended parallel robot. IEEE Trans. Robot. 30(3), 728–736 (2014). https://doi.org/10.1109/TRO.2013.2292451 9. Karimi Eskandary, P., Angeles, J.: The dynamics of a parallel Sch¨ onflies-motion generator. Mech. Mach. Theory 119, 119–129 (2018) 10. Karimi Eskandary, P., Belzile, B., Angeles, J.: Trajectory-planning and normalizedvariable control for parallel pick-and-place robots. J. Mech. Robot. 11(3), 031001 (2019). https://doi.org/10.1115/1.4042631 11. Lee, C., Lee, P.: Isoconstrained mechanisms for fast pick-and-place manipulation. In: Lou, Y., Li, Z. (eds.) Geometric Methods in Robotics and Mechanism Research: Theory and Applications, pp. 95–112. LAP Lambert Academic Publishing, Saarbr¨ ucken (1990) 12. Lee, C.C., Herv´e, J.M.: Type synthesis of primitive Schoenflies-motion generators. Mech. Mach. Theory 44(10), 1980–1997 (2009). https://doi.org/10.1016/j. mechmachtheory.2009.06.001 13. Makino, H., Kato, A., Yamazaki, Y.: Research and commercialization of SCARA Robot-The case of industry-university joint research and development. Int. J. Automat. Technol. 1, 61–67 (2007) 14. Mises, R.V.: Motorrechnung, ein neues Hilfsmittel der Mechanik. ZAMM – J. Appl. Math. Mech. / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik 4(2), 155– 181 (1924). https://doi.org/10.1002/zamm.19240040210 15. Pellicciari, M., Berselli, G., Leali, F., Vergnano, A.: A method for reducing the energy consumption of pick-and-place industrial robots. Mechatronics 23, 326–334 (2013)

Workspace Analysis and Torque Optimization on a SMG

457

16. Pierrot, F., Company, O., Shibukawa, T., Morita, K.: Four-degree-of-freedom parallel robot, US Patent 6,516,681 (2003) 17. Schreiber, L.T., Gosselin, C.M.: Sch¨ onflies Motion PARAllel Robot (SPARA): a kinematically redundant parallel robot with unlimited rotation capabilities. IEEE/ASME Trans. Mechatron. 24(5), 2273–2281 (2019) 18. Woo, S.H., Kim, S.M., Kim, M.G., Yi, B.J., Kim, W.: Torque-balancing algorithm for the redundantly actuated parallel mechanism. Mechatronics 42, 41–51 (2017). https://doi.org/10.1016/j.mechatronics.2017.01.002 19. Wu, G., Lin, Z., Zhao, W., Zhang, S., Shen, H., Caro, S.: A four-limb parallel Sch¨ onflies motion generator with full-circle end-effector rotation. Mech. Mach. Theory 146, 103711 (2020). https://doi.org/10.1016/j.mechmachtheory.2019.103711

Analysis of Running Expansion with Trunk and Pelvic Rotation Assist Suit by Using SLIP Model Hongyuan Ren1(B) , Takayuki Tanaka1 , Kotaro Hashimoto1 , and Akihiko Murai2 1 Graduate School of Information Science and Technology, Hokkaido University,

14-9 Kita-ku, Sapporo, Hokkaido 060-0814, Japan [email protected] 2 AIST/JST, PRESTO, 6-2-3 Kashiwanoha, Kashiwa, Chiba 277-0882, Japan

Abstract. In this paper, we evaluated the effect of the trunk and pelvic rotation assist suit in assisting human running motion. During running motion, when the angular momentum of the upper body and the lower body is closer to balance, the running efficiency will be improved. According to this fact, we developed the trunk and pelvic rotation assist suit to apply external forces in the upper body to balance the angular momentum and improve the running efficiency. To evaluate the assist efficiency of our assist suit, we used the spring-loaded inverted pendulum model (SLIP model) and conducted a running expansion experiment. We combined the results of SLIP model and experimental data to evaluate the effect of the assist suit by comparing the energy and athletic performance between natural and expanded running motion. Finally, we confirmed the trunk and pelvic rotation assist suit does have an assisting effect on the human running motion. Keywords: Running expansion · Spring-loaded inverted pendulum model · Power assist suit

1 Introduction Walking or running motion is the most basic and simplest motion for our humans. However, for those who have disorders in walking and lower physical strength seniors, walking or running motion become a difficult motion for them. Especially now, the population aging problem has become a serious social problem in many countries. Our body functions will decline as aging, appears as lower metabolic rate and decline muscle areas and strength [1], which means even a simple task will become more difficult for them. Here, we focus on the walking and running motion, built a wearable power assist suit to reduce the burden, balance the angular momentum and the energy of the upper body and the lower body, and enhance the efficiency during walking or running motion [2]. Different from these power assist suits which are equipped on lower limbs to assist walking motion, such as the HAL power assist system [3], we focus on the upper body system. Walking or running motion does not only based on the movement of our legs, but it is a complicated motion that needs our entire body to participate in. Our spine as © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 458–466, 2021. https://doi.org/10.1007/978-3-030-58380-4_55

Analysis of Running Expansion with Trunk and Pelvic Rotation

459

the link of the upper body and the lower body, it plays an important role of transmitting the energy and cancel the angular momentum generated between the upper body and the lower body while walking or running, and makes more contributions to the gait, which is called spinal engine theory [4]. According to this, we built the trunk and pelvic rotation assist suit, provide a rotation force to the upper body for assisting walking and running motion. As we built the system, we want to evaluate the effect of the system to figure out how it affects human motion. But human motion is very complicated and has many degrees of freedom (DOFs), it is very difficult to analyze. To solve these problems, in this paper we used spring-loaded inverted pendulum model (SLIP model) [5]. It is a simple model with a spring and a mass and has already been proved in robot design and control fields [6]. By using this model, the complex human motion mechanisms can be represented parametrically, and extremely reduce the DOFs to 1, make it easy to analyze. Instead of the original two-dimensional SLIP model, here we used a three-dimensional SLIP model [7] to make it more accurate and comprehensive to represent the human motion, and it is more useful to evaluate our trunk and pelvic rotation assist suit. This paper is for analysis of running expansion with trunk and pelvic rotation assist suit by using SLIP model. In Sect. 2, we describe the details of our trunk and pelvic rotation assist suit, including the structure and the principle. In Sect. 3, we describe the method of the three-dimensional SLIP model, which we used to analyze human motion. In Sect. 4, we describe the running expansion experiment and the result analysis, also comprise result discussion. We conclude in Sect. 5.

2 Trunk and Pelvic Rotation Assist Suit Walking or running motion is not the pure motion of the legs, it requires coordination throughout the body. Normally, when we keep walking or running, we will swing our arms or trunk involuntarily, when we move our legs, there is a balance between our upper body and lower body to keep us stable [8]. For our human beings, spine as the link of the lower body and the upper body, not only supporting and protecting the upper body and the head but also transmitting the energy and canceling the angular momentum generated between the upper body and the lower body, like an engine drive us to keep walking or running, which has been proposed by Gracovetsky et al. [4]. According to this theory, we built our trunk and pelvic rotation assist suit depicted in Fig. 1, it provides an external torque to the upper body for improving movement efficiency and declining energy loss by smoothly transferring energy between the trunk and the pelvis. Figure 1a depicts the structure of the trunk and pelvic rotation assist suit designed in CAD, consists of the motor as a power source, the flexible shaft as the link between the pelvis and the trunk, transfer the power between them, and due to the high flexibility of the shaft, it will not excessive impact the motion of the wearer. Two metal plates are respectively equipped on the chest mount and pelvis mount to secure the moment arm, the output torque of the motor will be transferred to the trunk by the flexible shaft to create a rotary motion between the trunk and the pelvis, also a sliding mechanism is provided on the chest mount to reduce the effect of the assist suit on the forward flexion motion of the trunk. The control system of the assist suit is periodic input control. It is a useful method

460

H. Ren et al.

for assisting the walking or running motion, which is a periodic motion, and is suitable for improving the gait of the wearer. It is a method to dynamically stabilize a mechanical system with a periodic motion to an arbitrary energy state, and by inputting a periodic signal to the controller to obtain an output that is synchronized with the periodic input frequency. Here, we used the right hip joint angle which is measured by IMU sensor as the input to obtain the target trunk rotation angle which is synchronized with the motion of leg as the output to control the assist suit. Moreover, by changing the deformation degree of the controller, it will affect the phase of the output waveform, therefore, it is possible to control the energy of the system by advancing or delaying the phase of the periodic motion of the input. Figure 1b shows the photo of the trunk and pelvic rotation assist suit. It has 412 mm length, 340 mm width, 85 mm height, and the weight is 3.0 kg. The motor power is 100 W, the reduction ratio is 72:1, and has a 171 rpm rated speed. The rated output shaft torque of our assist suit is 3.29 Nm, and the operating frequency is 50 Hz.

(a) CAD structure design.

(b) Wearing assist suit.

Fig. 1. Trunk and Pelvic Rotation Assist Suit. (a) structure of the trunk and pelvic rotation assist suit designed in CAD, (b) photo of the trunk and pelvic rotation assist suit.

3 Spring-Loaded Inverted Pendulum Model In this section, we describe the model which we used to analyze human running motion and evaluate the trunk and pelvic rotation assist suit. Spring-Loaded Inverted Pendulum model (SLIP model) is a 1 DOF model consist of a mass point and a spring, proposed by Blickhan et al. [5]. It extremely reduces the DOFs of human motion, simplifies the complicated motion to make it representable and analyzable. It is a successful theory that has been widely used in robot fields. The SLIP model we used was proposed by Seipel et al. [7], it can represent the human motion in three-dimensional, compared with two-dimensional SLIP model, obviously, it is a more accurate and comprehensive model. We used this model to represent human motion approximately, and by analyzing the parameters of the model to evaluate the effect of the assist suit. Figure 2 shows the running motion represented by the three-dimensional SLIP model. It consists of a mass point, which means the center of mass (COM) of the human, and a massless and passive spring leg. The running motion can be separated into two phases, respectively, stance phase and flight phase. Stance phase occurs when the foot is on

Analysis of Running Expansion with Trunk and Pelvic Rotation

461

the ground and the spring leg of the model is compressed and storing elastic potential energy. Besides, flight phase occurs when both feet are in the air and the spring leg is in a resting position. Touchdown (TD) and Liftoff (LO) are the critical states between the translation of these two phases. The critical state between the previous flight phase and the next stance phase is TD, LO is the link from stance phase to flight phase. A single gait consists of TD, stance phase, LO and flight phase, by cycling them to keep running.

Fig. 2. The three-dimensional SLIP model.

Fig. 3. The coordinates of the SLIP model.

The coordinate systems of the three-dimensional SLIP model are shown in Fig. 3. The Cartesian coordinate (x w , yw , zw ) represents the world coordinate, and (x h , yh , zh ) represents the human coordinate. We consider the center of mass as the origin of the human coordinate, and we can get the transformation equation between the world coordinate and the human coordinate as:    h  P I3×3 Pcw Pw = (1) 0 1 1 1       w w T , Pw = Xw Yw Zw T , Ph = Xh Yh Zh T . Pcw = Xw (2) c Yc Zc where Pcw means the position of the COM in the world coordinate, Pw is the position in the world coordinate, and Ph is the position in the human coordinate. |L(t)| is the length of the spring leg which changes over time during human movement, the formula can be written as:   L(t) = xFh (t), yFh (t), zFh (t) (3) where xFh (t) is the x-component of the position of the force acting point in the human coordinate, so does yFh (t), zFh (t). According to this, by using Lagrange’s equations, the equation of motion of the SLIP model can be exported as:  ¨ = k L(t) L0 − 1 − gez L(t) (4) |L(t)| m ¨ = −gez L(t)

(5)

where Eq. (4) describes the motion of stance phase, and Eq. (5) represents the motion of flight phase. ez is the unit vector of z-direction. By using this model, we can analyze running motion simply, and the evaluation of the trunk and pelvic rotation assist suit will be introduced in the next section.

462

H. Ren et al.

4 Running Expansion Experiment and Result Analysis 4.1 Running Expansion Experiment We performed a running expansion experiment to evaluate the effect of our assist suit. Subject running motion measurement is based on motion capture system (VICON), electromyography (EMG) is used to detect the electrical activity produced by the skeletal muscles of the subject, reflect the burden on body during the running motion, the force plate mounted on the ground used to measure the reaction force generated by the ground. The experimental environment is indoors, on the flat ground. The subject is a male (1.67 m height and 60.0 kg weight), running in natural status and assistance status, but here to keep the same weight we consider the natural status as wearing assist suit, but the power was not activated, and in the assistance status the power of the assist suit was activated. Respectively measured 20 times under their own experimental conditions. The running intensity was not hard, just keep a smooth-running status. 4.2 Result Analysis We measured 20 times in two different running status. And got 28 effective steps in natural status and 24 effective steps in assistance status. To unify the result, we analyzed the 22 effective steps in each status.

Fig. 4. Muscles electrical activities.

Fig. 5. Muscles position.

Figure 4 depicts the electrical activities of the subject muscles measured by EMG. The muscles are External oblique, Erector spinae, Gluteus maximus, Quadriceps femoris, Biceps femoris, successively. The position of these muscles is shown in Fig. 5. As we can find that, both gluteus maximus and the right biceps femoris activities are significantly declined as the effect of the assist suit, though in other muscles this phenomenon does not be observed. It means our trunk and pelvic rotation assist suit does have some effect on human running motion to decline the burden of the muscle and assist the wearer to keep running motion. In order to analyze the effect more specifically, we used the SLIP model. We referred the SLIP model parameters identification method from Murai et al. [9]. The SLIP model parameters identification equation is as: Ef =

T t

(|F(t)| − k(L0 − |L(t)|))2

(6)

Analysis of Running Expansion with Trunk and Pelvic Rotation

463

k and L0 are the SLIP model parameters. k means the spring constant of the spring leg and L0 is the resting length of the spring leg. F(t) is the ground reaction force measured by the force plate. By minimizing Eq. (6) using a nonlinear programming solver we can obtain those parameters, and the results are shown in Table 1. The nonlinear programming solver we used in this paper is the generalized reduced gradient method. It is one of the most popular methods to solve nonlinear optimization problems. The variables are separated into two categories, dependent and independent, and by iterative computing the reduced gradient until to find the optimized value. Table 1. SLIP-model parameters k (N/m) L0 (m) Natural

9582

1.01

Assistance 9008

1.01

The spring constant k is declined in assistance status compared with natural status, obviously due to the assist suit, which means our suit equipped on the upper body does affect the motion of the lower body. By using these parameters, we compared the force and energy to analyze the changes that happen to the spring constant k. First, the force of the spring leg, in this model, it represents the force that is provided by the human. The force compared results are shown in Fig. 6. The equation is as: |FSLIP (t)| = k(L0 − |L(t)|)

(7)

Second, the energy of the spring leg, the potential energy Ep is expressed as the elastic potential energy stored in the spring, and the system energy also contains the kinetic energy Ek . The energy compared results are shown in Fig. 7. Ep and Ek can be represented as: 1 k(L0 − |L(t)|)2 2 1  ˙ 2 Ek (t) = mL(t) 2

Ep (t) =

Fig. 6. SLIP model spring leg force compare.

(8) (9)

464

H. Ren et al.

Fig. 7. SLIP model spring leg energy compare. Table 2. SLIP model force and energy compare Natural

Assistance

Average Peak F (SD) [N] 1169 (72)

1168 (90)

Ep (∗) [J]

46032 (17.6%)

55506 (22.9%)

Ek (∗) [J]

215649 (82.4%) 187177 (77.1%)

ETotal [J]

261681

242683

* The proportion of the energy in total energy

4.3 Discussion The specific numerical comparison can be found in Table 2, in this part, we make a discussion on the result. It is focused on the energy changes and athletic performance between natural and assistance status, and the data compare is shown in Table 3. Table 3. Energy and the athletic performance compare Natural (SD)

Assistance (SD)

Each step consumes average energy [J]

11894 (1373)

11031 (1450)

Running velocity [m/s]

2.48 (0.16)

2.16 (0.17)

Stride length [m]

1.93 (0.07)

1.71 (0.10)

Running cycle [s]

0.75 (0.04)

0.83 (0.02)

To verify the accuracy of the SLIP model, we simulated the running motion, the simulation result was closed to the real experiment data, in natural running status, the running speed is 2.48 m/s measured by sensors, and the simulation result of running speed is 2.45 m/s. Compared with natural status, a large change in force is not observed, but the potential energy is increased, and the kinetic energy is declined. Each step consumes average energy of these two statuses is closer, but the energy in natural status is greater than that in assistance status. In natural status, the energy is provided only by the human, but in assistance status, the energy is provided by the human and the power assist suit. Therefore, we can obtain that the energy which should be provided by the human is declined while running, as the effect of the assist suit. Which Fig. 4 also proves this

Analysis of Running Expansion with Trunk and Pelvic Rotation

465

suppose, the activities of both gluteus maximus and the right biceps femoris muscle have obviously declined, which means the burden of human was reduced. The muscle activities were measured by EMG sensor as a phenomenon of the experiment, and the result was analyzed by using SLIP model, both show the energy which should be provided by the human is declined while running, proved that our suit does have an assistance effect to human motion. Besides, the other result, the differences between the spring constant k in the SLIP model, it affects the conversion between the potential energy and the kinetic energy, the reduction of k declined the conversion efficiency, led to an increase in the proportion of potential energy, compared with natural status. The athletic performance can be described in three aspects, running velocity, running cycle and stride length. Compared with natural status, the velocity and the stride length are declined, but the cycle is increased, which means gait becomes slower. According to the potential energy is increased but the spring constant k is declined in SLIP model compared with natural status, which means the deformation of the spring leg become lager, in other words, the undulate of human becomes more obvious during running motion compared with natural status, which we are not expected.

5 Conclusion In this paper, we used a three-dimensional SLIP model to evaluate the effect of the trunk and pelvic rotation assist suit based on a running expansion experiment. Our trunk and pelvic rotation assist suit based on spinal engine theory and focused on the upper body to balance the energy between the upper body and lower body during the running or walking motion, help to keep an efficient gait while running or walking. By combing the result analyzed by SLIP model with the data of EMG sensor, proved that our assist suit does have an assisting effect on human running, appears in reducing the energy which should be provided by the human during the running motion, and reduce the burden of human. However, the athletic performance while wearing the assist suit did not show the expected results, which we will focus on this problem in our future work, and since we have proved the assistance effect of our assist suit, we will continue ameliorating the mechanical design to make it more comfortable to wear and optimizing the control algorithm to improve the assistance effect, keep improving our assist suit to help the wearer keep a more efficient gait during running or walking motion.

References 1. Nair, K.S.: Aging muscle. Am. J. Clin. Nutr. 81(5), 953–963 (2005) 2. Hashimoto, K., Tanaka, T., Kusaka, T.: Walking assistance and resistance of walking motion by trunk and pelvis motion assist. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 8597–8602. IEEE, Madrid, Spain (2018) 3. Kawamoto, H., Sankai, Y.: Power assist system HAL-3 for gait disorder person. In: 8th International Conference on Computers for Handicapped Persons, pp. 196–203. Springer, Linz, Austria (2002) 4. Gracovetsky, S.A., Iacono, S.: Energy transfers in the spinal engine. J. Biomed. Eng. 9(2), 99–114 (1987)

466

H. Ren et al.

5. Blickhan, R.: The spring-mass model for running and hopping. J. Biomech. 22(11–12), 1217– 1227 (1989) 6. Saranli, U., Koditschek, D. E.: Template based control of hexapedal running. In: 2003 IEEE International Conference on Robotics and Automation (Cat. No. 03CH37422), vol. 1, pp. 1374– 1379. IEEE, Taipei, Taiwan (2003) 7. Seipel, J.E., Holmes, P.: Running in three dimensions: analysis of a point-mass sprung-leg model. Int. J. Robot. Res. 24(8), 657–674 (2005) 8. Lafage, V., Schwab, F., Skalli, W., et al.: Standing balance and sagittal plane spinal deformity: analysis of spinopelvic and gravity line parameters. Spine 33(14), 1572–1578 (2008) 9. Murai, A., Tada, M.: Multilayered kinodynamics simulation for detailed whole-body motion generation and analysis. In: 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 581–587. IEEE, Brisbane, Australia (2018)

Trajectory Planning Strategy for Multidirectional Wire-Arc Additive Manufacturing Markus Schmitz(B) , Carlo Weidemann, Burkhard Corves, and Mathias H¨ using Institute of Mechanism Theory, Machine Dynamic and Robotics, RWTH Aachen University, 52074 Aachen, Germany [email protected] http://www.igmr.rwth-aachen.de

Abstract. Robotic systems in additive manufacturing (AM) processes offer new possibilities to meet constantly growing requirements for product individualization and small batch sizes. Solids are produced by the layered addition of material, which represents a benefit compared to conventional processes, especially in the production of complex geometries with cavities and undercuts. Recently, research into the entire process of AM from CAD models to the execution of an optimized trajectory has moved into focus. This is motivated by the use of Multidirectional Additive Manufacturing (MDAM) for Wire Arc Additive Manufacturing (WAAM). Since the material is fed eccentrically, the freedom of movement of the welding torch is strongly restricted. The novel procedure of pure manipulation of the component (with six degrees of freedom) can therefore avoid this problem and obtains optimization potentials. This paper gives an overview of the characteristics of MDAM for WAAM processes. Besides an introduction to the extended MDAM process chain, a method for optimizing trajectories is presented and analyzed. To improve the welding result, orientation tolerances are applied to the unique path supporting poses along the welding path. Additionally, implementing a custom cost function for the trajectory optimization has a positive impact on the acceleration and jerk levels. The generated trajectories thus follow the specified welding path with reduced acceleration and jerk. Keywords: Wire Arc Additive Manufacturing · Multidirectional Additive Manufacturing · Additive manufacturing · Trajectory planning · ROS · Descartes

1

Multidirectional Additive Manufacturing Using WAAM

Additive manufacturing processes for metallic materials are becoming increasingly important in the context of resource-saving production and expanded c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 467–475, 2021. https://doi.org/10.1007/978-3-030-58380-4_56

468

M. Schmitz et al.

lightweight construction strategies and are the focus of current research and development worldwide. Future strategies for modern, highly individualized manufacturing processes, especially in high-wage countries, regard AM processes as a key technology with major development potential [6]. Tungsten inert gas (TIG) welding or the related plasma welding are the main methods used for welding sensitive high-performance materials. Due to their high process stability, very precisely adjustable welding parameters and the decoupling of energy and material input, these processes are well suited for processing highly reactive alloys or alloys susceptible to hot cracking [5]. A potential area of application for WAAM is the near-net-shape production of components from materials that are difficult to process and/or expensive. It allows buy-to-fly ratios (BTFR, ratio of the weight of the semi-finished material to the weight of the finished part) to be significantly reduced compared to the previous method of machining from a solid block. Exemplary studies by Cranfield University show reductions in the BTFR from 45 to 2.9 and lower, which corresponds to material savings of over 93% [3]. In addition to the known layer-by-layer build-up methods in one buildup direction, additive manufacturing methods are increasingly being extended by additional degrees of freedom. Thereby, additional degrees of freedom can be introduced into the substrate plate as well as into the printing or welding torch. MDAM represents the special case where only the substrate plate is manipulated under a fixed welding torch.

Fig. 1. WAAM system with manipulator (M A), substrate holder (SH), substrate plate (SP ) and stationary welding torch (W T ).

MDAM offers many advantages for the additive processing of different materials and applications. For example, the conventional plastic 3D printing can be extended by this concept. In a WAAM process, as shown in Fig. 1, the electric light arc burns between a non-melting electrode and the substrate plate (SP ). The filler material is fed

Trajectory Planning Strategy for Multidirectional WAAM

469

into the welding zone, which is melted in the arc, dripped off and solidified on the substrate plate (SP ). This plate is held by the substrate holder (SH). Both parts represent the end effector of the manipulator (M A). The welding torch (W T ) necessary for wire feeding and arc generation is stationary. The substrate is applied to the platform during the welding process. The defined feeding direction of the wire with respect to the welding path requires a unique and full spatial description of the path poses. Consequently, when using a 6-axis industrial robot, no redundancy is maintained. Increasing the robot’s degrees of freedom would be one way of providing new optimization space through redundancy. However, high availability and low costs motivate the use of 6-axis industrial robots. The arrangement of the components to each other influences both the reachability of all necessary robot poses along a given welding path and the kinematic and dynamic properties when executing a trajectory. This suitable arrangement is not the subject of this paper and has already been made for all the tests carried out. The fixed definition of the welding direction by feed wire or sensors results in a defined direction of movement of the end effector in relation to the welding torch or the world reference frame. Thus, the poses along a welding path are firmly defined. Without a suitable trajectory planner, it is possible that poses on the path are not followed on the direct path and, for example, robot configuration needs to be changed during the welding process. This in turn leads to high undesired speed and furthermore makes the execution of an additive manufacturing process impossible. In addition, a trajectory planner should follow a defined path between the support poses while maintaining a defined speed. The occurring problems motivate the development of a trajectory planner, which on the one hand travels the geometrically discretized space in R3 on a direct path and on the other hand minimizes oscillations around the path. In order to reduce jerk and acceleration in particular, the orientation around the vertical axis of the welding torch should be released within narrow limits and the default cost function should be changed from minimizing the transition angles to minimizing the joint velocities. The influence of this release should be analyzed especially for straight paths and curves of different form.

2

Trajectory Planning Strategy for MDAM

Trajectory planning in MDAM with robots means the extension of a path by a time component and information for the welding process. The input consists of welding paths support poses, which have been geometrically planned in the workpiece reference frame. The trajectory planner has the possibility to modify the path support poses for the best possible welding result. Furthermore, in the context of this publication, an optimization within the scope of orientation variations and changing the default cost function shall ensure a clean execution of the trajectory. The clean execution of a welding path is quantified by small positional deviations from the desired welding path, rotational deviations from the prescribed wire feeding direction, high continuity of the welding velocity, low acceleration and jerk in the robot joints.

470

M. Schmitz et al.

The deviation from the prescribed wire feed direction is the optimization space in trajectory planning using the Descartes package within ROS [10]. Descartes is a planning library for semi-constrained trajectories, which creates well behaved plans minimizing joint motions. Furthermore, it can handle hybrid trajectories consisting of joint, cartesian and specialized poses. Its main use case is robotic routing, blending and welding with negligible orientation about the tools Z-Axis [9].

Fig. 2. Integration of Descates and ROS into the MDAM Process

Searching for valid robot configurations and generating a solution is carried out by the Descardes planner package. It contains the dense and the sparse planner of Descartes. In this paper the dense planner is used, because it accounts for all trajectory poses. The procedure for generating a solution is as follows: For every trajectory pose and its given tolerances valid inverse kinematic solutions are calculated with respect to the search discretization. The default cost function computes a maximum joint angle (see Eq. 2) from the velocity limits of two

Trajectory Planning Strategy for Multidirectional WAAM

471

consecutive poses. After this, the absolute deviation of each joint is calculated and accumulated (see Eq. 1, 4), if the deviation is smaller than maximum angle (see Eq. 3). Then the results of those calculations are assembled into a planning graph. The inverse kinematic solutions form the nodes and the movement costs form the edges of the planning graph. At the end, the Dijkstra algorithm is applied to find the robot configurations along the specified path with the lowest transition cost. Instead of using the default cost function Descartes offers the possibility to define custom cost functions to accumulate joint velocities (see Eq. 5). An overview of the integration of Descates and ROS into the presented MDAM Process is shown in Fig. 2. −−−−→ −−−−→ → − ΔΦi = abs(Φi,start − Φi,stop ) −−→ → − − v− i,limit Δ θi = Δt → − → − Δ θi ≥ ΔΦi → − costdef ault = accumulate(ΔΦi ) → − ΔΦi costcustom = accumulate( ) Δt

TCP Path

30

Y [mm]

(3) (4) (5)

Print Path target actual

20

20

10

10

0

0

-10

-10

-20

-20

470

(2)

30

target actual

-30 460

(1)

480

490

500

X [mm]

510

520

530

540

-30 -40

-30

-20

-10

0

10

20

30

40

X [mm]

Fig. 3. TCP path in base frame (left). Welding path in substrate plate frame (right). Blue cross: support pose; red line: actual robot path

The trajectory package contains three ways to define a trajectory pose and assigns tolerances and timing: Joint trajectory, cartesian trajectory and axial symmetric poses [10]. In this paper only cartesian trajectory poses are used, because the major path-requirements in addtive manufacturing. Those cartesian poses can be constructed with four different transformations: Two fixed transforms from the manipulator (M A) base to the welding torch (W T ) or from the manipulator’s wrist to the substrate holder (SH). And two under-constrained

472

M. Schmitz et al.

transforms from the welding torch to the effective point on the welding torch or from the substrate holder to the effective point on the substrate plate (SP ). Those possibilities effect two different ways to specify the path support poses: TCP poses with rotational tolerances around the substrate plate or welding poses with rotational tolerances around the welding torch. In this paper the latter is used, because it results in a trajectory, which lies on the specified welding path (Fig. 3). The former causes an angled welding path, because the tolerances are applied on the TCP poses. Table 1. Tested DoE stages for the optimization of multidirectional welding trajectories. Variables

Levels of variables

Radius

Sharp edges Smoothed edges

Discretization

Rough

Orientation tolerance 0◦

Medium

Fine

±1◦

±5◦

In addition, the initial redesign of the input path could also have an effect on the executed trajectory considering the transformation from welding path to TCP path. For example, a smoothing of sharp geometric segments in the path can affect the generated joint accelerations and the position accuracy. In addition, the discretization of path support poses can have a positive effect on position accuracy. However, it must be taken into account that the planner is thereby deprived of optimization space, which in turn can have negative effects. The effects of different planning parameters on the trajectory were systematically tested using Design of Experiments (DoE, see Table 1).

3

Results

Figure 4 shows the welding path of three characteristic test conditions ([sharp edges, rough discretization, no orientation tolerance (0◦ )], [smoothed edges, rough discretization, orientation tolerance (±5◦ )] and [smoothed edges, fine discretization, orientation tolerance (±5◦ )]. It can be seen that the positional accuracy only is relevant in curved segments. In general, the position accuracy increases with the change from sharp to smoothed edges, a higher discretization of the input path and rotational freedom around the welding torch axis. The influence of a custom cost function is not considered. Releasing the orientation constraint around the welding torch axis has a positive effect on the trajectory execution resulting in lower acceleration and jerk levels. On the contrary it has a not favourable effect in combination with both cost functions. The manipulator always moves in a configuration with maximal or minimal allowed tolerance (see Fig. 5). Yet these deviations from the prescribed

Trajectory Planning Strategy for Multidirectional WAAM R=false F=1 O=0

40 30

R=true F=1 O=5

40

target actual

30

20

10

10

0

0

0

-10

-10

-10

-20

-20

-20

-30

-30

-30

-40 -30

-40 -30

X [mm]

20

10

-10

0

10

20

30

-20

-10

Y [mm]

target actual

30

20

-20

R=true F=5 O=5

40

target actual

0

10

20

-40 -30

30

473

-20

-10

Y [mm]

0

10

20

30

Y [mm]

Fig. 4. Three characteristic tests in substrate plate (SP ) frame: Sharp edges, rough discretization, no orientation tolerance (0◦ ) (left). Smoothed edges, rough discretization, orientation tolerance (±5◦ ) (middle). Smoothed edges, fine discretization, orientation tolerance (±5◦ ) (right) Deviation of Euler Angle

20

Deviation [deg]

O=0 O=5

10 0 -10 -20

0

10

20

30

40

50

60

Time [sec]

Fig. 5. Deviation of Euler Angle around Welding Torch Axis Acceleration of Joint 6

4

0 -2 -4 18

19

20

21

Time [sec]

22

23

24

default custom

40

2

17

Jerk of Joint 6

60 default custom

Jerk [rad/sec³]

Acceleration [rad/sec²]

6

25

20 0 -20 -40 17

18

19

20

21

22

23

24

25

Time [sec]

Fig. 6. Detail of a curved segment: acceleration (left) and jerk of joint six (right)

wire feed direction are within a tolerable range considering that the welding process is not effected by small rotational changes. Varying the parameters presented in Table 1 does not significantly affect the mean welding velocity, but its standard deviation. Small values indicate a low fluctuation of the welding velocity. Fluctuations only occur during curved segments. Using smoothed edges, a rough discretization and maximal rotational tolerances of ±5◦ with the custom cost function is showing the highest continu-

474

M. Schmitz et al.

ity of the welding velocity. As mentioned above rotational freedom enhances the acceleration and jerk levels. In addition, smoothed edges, rough discretization are beneficial for the dynamics, too. The custom cost function has a positive impact, as shown in Fig. 6.

4

Conclusion

All in all, using smoothed instead of sharp edges is always beneficial. A rough discretization benefits the continuity of the welding velocity, acceleration and jerk levels but leads to high positional deviations. The custom cost function is promising, but can have a bad impact on the trajectory execution in special cases. Figure 7 shows the experimental results. A elliptical half-shell was printed with the help of Welding and Jointing Institute at RWTH Aachen University. These results and findings motivate and require further investigation.

Fig. 7. Results of an elliptical half-shell, additive manufactured at the test setup of Welding and Joining Institute at RWTH Aachen University

References 1. Detert, T., Lorenz, M., Schmitz, M., H¨ using, M., Corves, B.: Robotergef¨ uhrte objektmanipulation f¨ ur die generative fertigung. In: Beitelschmidt, M. (ed.) Tagungsband 12. TUDpress, Dresden, Kolloquium Getriebetechnik. Studientexte zur Dynamik und Mechanismentechnik (2017)

Trajectory Planning Strategy for Multidirectional WAAM

475

2. Ding, D., Pan, Z., Cuiuri, D., Li, H.: Process planning for robotic wire and arc additive manufacturing. In: 2015 IEEE 10th Conference on Industrial Electronics and Applications (ICIEA), pp. 2000–2003. IEEE (2015) 3. Frazier, W.E.: Metal additive manufacturing: a review. J. Materials Eng. Performance 23(6), 1917–1928 (2014) 4. Gebhardt, A.: Additive Fertigungsverfahren: Additive Manufacturing und 3DDrucken f¨ ur Prototyping-Tooling-Produktion. Carl Hanser Verlag GmbH Co KG (2017) 5. Reisgen, U., Stein, L.: Grundlagen der F¨ ugetechnik: Schweißen, L¨ oten und Kleben, Fachbuchreihe Schweißtechnik, vol, vol. 161. DVS Media GmbH, D¨ usseldorf (2016) 6. Richter, S., Wischmann, S.: Additive fertigungsmethoden: Entwicklungsstand. IITBerlin (2016) 7. Santos, E.C., Shiomi, M., Osakada, K., Laoui, T.: Rapid manufacturing of metal components by laser forming. Int. J. Mach. Tools Manuf. 46(12–13), 1459–1468 (2006) 8. Schmidt, T.: Potentialbewertung generativer Fertigungsverfahren f¨ ur Leichtbauteile. Springer (2016) 9. Southwest Research Institute: ROS-industrial advanced developer’s training - session 5 (2017) 10. Lamoine, V.: Ros wiki - descartes (2018). http://wiki.ros.org/descartes

Stiffness Modeling of Planar Robotic Manipulators: Model Reduction and Identifiability of Parameters Shamil Mamedov1(B) , Dmitry Popov1 , Stanislav Mikhel1 , Alexander Klimchik1 , and Anatol Pashkevich2,3 1

3

Innopolis University, Universitetskaya St, 1, Innopolis, Tatarstan 420500, Russia {sh.mamedov,d.popov,s.mikhel,a.klimchik}@innopolis.ru 2 IMT Atlantique Nantes, 4 rue Alfred-Kastler, 44307 Nantes, France [email protected] Laboratoire des Sciences du Num´erique de Nantes (LS2N), UMR CNRS 6004, 1 rue de la Noe, 44321 Nantes, France

Abstract. The paper deals with stiffness modeling of planar robotic manipulators and calibration of the model parameters. It was proved that the conventional stiffness models, which are widely used in practice, are parametrically redundant and only limited subset of link/joint compliance matrices elements can be estimated. To overcome this difficulty, several model reduction techniques were developed and corresponding parameter estimation methods were proposed. The validity and efficiency of these methods were evaluated via relevant simulation study. Keywords: Planar manipulators · Stiffness modeling · Model reduction · Parameter identifiability · Estimation techniques

1

Introduction

Many modern anthropomorphic robots, which are aiming to approach human behavior, are based on the highly compliant manipulating mechanisms. In contrast to classical robots with rigid transmissions and links, they are more suited for accomplishing complex tasks and working in cooperation with humans. However, the actuation techniques for such robots are more complex compared with their rigid counterpart, because relevant control algorithms must incorporate both traditional kinematic model describing the manipulators geometry and also its non-negligible elasto-static model describing the manipulators compliance/stiffness with respect to the external wrench. To model the manipulator elasticity, currently there exist three main techniques currently, that are: the Finite Element Analysis (FEA), the Matrix Structural Analysis (MSA), and the Virtual Joint Method (VJM). The most accurate of them is the FEA, which allows modeling links and joints with their true dimension and shape but is not suitable for real-time control because of very c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 476–485, 2021. https://doi.org/10.1007/978-3-030-58380-4_57

Stiffness Modeling of Planar Robotic Manipulators

477

high computational expenses [1–3]. The MSA can be treated as certain compromise between the model accuracy and complexity. It incorporates the main ideas of the FEA, but operates with rather large compliant elements such as beams, arcs, cables, etc. This obviously leads to some reduction of the computational efforts but it is still insufficient to be used in real time [4,13,14]. Finally, the VJM method – the most attractive for control applications – is based on the extension of the traditional rigid model by adding the virtual joints (localized springs), which describe the elastic deformations of the links, joints and transmissions [6,7,12,18,20]. This technique provides rather simple analytical model that allows fast computation of the end-effector Cartesian stiffness or compliance matrix, which is expressed in the form of the weighted sum of some parameters describing elasticity of separate links and joints. Because of its simple structure, the VJM-based models find the widest application in practical robotics but the issues related to the parameter estimation for such models still create difficulties for robotic engineers. At present, there are two main techniques to find parameters of the VJMbased model, a numerical and experimental one. The first of them assumes that 3D-descriptions of the manipulator components are available and accurate enough, so that the desired parameters are obtained straightforwardly from FEA modeling of each link and joint (considering them separately and independently from the manipulators configurations) [9,17]. However, in practice the accuracy of such technique is limited. More reliable parameter estimates are obtained by experimental data processing that deals with the simultaneous least-square solution of the several systems of linear equations corresponding to different manipulator configurations [10,16,19]. In spite of apparent simplicity, there are a number of difficulties related to the rank-deficiency of the aggregated regression matrix. In particular, in typical practical cases the entire set of the stiffness/compliance parameters is redundant with respect to the inputoutput relations described by considered model. As a result, the conventional least square solution based on the Moore–Penrose pseudo-inverse may produce mathematically correct but physically unacceptable results (with negative link stiffness, for instance). Hence, before used for the identification, the model must be reduced by eliminating or fixing certain parameters, which it is not easy and obvious. It is worth mentioning that some similar problems arise in calibration of robot geometric models, where there were introduced the notions of model completeness and irreducibility [5,8,15]. However, compared to this domain, the stiffness model calibration possesses some particularities that do not allow to apply directly the results from the robot geometry calibration. In particular, the considered class of input-output relations has completely different structure (corresponding equations simultaneously include both the kinematic Jacobians and their transposes). Besides, the model completeness problem does not exist here, while the model reduction is very specific and requires development of dedicated techniques, which are proposed in this paper.

478

2

S. Mamedov et al.

Problem Statement

Let us consider a general n-dof planar serial manipulator composed of relevant number of non-rigid links connected by motor-actuated joints, which also possess certain compliances. Assuming that the link and joint flexibilities are non-negligible, traditional geometric model of such manipulator (also known as the direct and inverse kinematics) must be extended by some relations allowing estimating the end-effector reaction to external forces and torques. In the most cases this reaction can be described by a linearized “force-deflection” relation similar to the Hook’s law but dealing with more general 3-dimensional entities w = K e (q) · Δt or Δt = ke (q) · w

(1)

where Δt ∈ R3 denotes the end-effector displacement (linear Δx, Δy and angular Δφ), w ∈ R3 is the external wrench (forces Fx , Fy and torque Mφ ) applied to it, K e (q) ∈ R3×3 and ke (q) ∈ R3×3 are respectively the stiffness and compliance matrices that depend on the current manipulator configurations defined by the joint coordinate vector q ∈ Rn . Because of their physical nature, the matrices K e and ke are assumed to be symmetrical and positive definite. In general, a straightforward computing of the configuration-dependent matrix K e (q) or its inverse ke (q) requires application of very time consuming FEA-based simulation. To reduce computational efforts, practical engineers use the following simplification all flexibilities are localized in manipulator joints and presented in the form of multidimensional virtual springs describing compliance of both the links and actuators. This approach allows expressing the entire manipulator compliance as a superposition of partial link/joint compliances in accordance with the following expression ke (q) =

n  

 J i (q) · ki · J i (q)T ,

(2)

i=1

where the constant matrices ki ∈ R3×3 incorporate compliance parameters of the virtual springs and the configuration-dependent Jacobian matrices J i (q) ∈ R3×3 describe influence of virtual springs deflections on the end-effectors displacement. In the Eq. (2), the Jacobian matrices can be easily computed using conventional techniques adopted for manipulators with rigid links, while compliance matrices ki can be identified from experimental data that consists of pairs (kje , q j ), j = 1, . . . , m using ordinary least squares. For that we transform Eq. (2) into the regressor form using the vectorization operator vec(·) and the Kronecker product ⊗ applied to the 3 × 3 matrices J ij  J i (q j ) ⎤⎡ ⎤ ⎡ ⎤ ⎡ vec(k1 ) J 1 (q 1 ) ⊗ J 1 (q 1 ) . . . J n (q 1 ) ⊗ J n (q 1 ) vec(k1e ) ⎦ ⎣ . . . ⎦ = ⎣ . . . ⎦ , (3) ⎣ ... ... ... J 1 (q m ) ⊗ J 1 (q m ) . . . J n (q m ) ⊗ J n (q m ) vec(kn ) vec(km e ) that contains 9m scalar equations for 9n scalar variables incorporated in the 3×3 compliance matrices k1 , . . . , ki . Further, taking into account the symmetricity

Stiffness Modeling of Planar Robotic Manipulators

479

of the matrices ke and k1 , . . . , ki , the number of equations and variables can be reduced down to 6m and 6n respectively, yielding the reduced linear system ⎤⎡ ⎤ ⎡ ⎤ ⎡ vech(k1 ) L (J 11 ⊗ J 11 ) D . . . L (J n1 ⊗ J n1 ) D vech(k1e ) ⎦ ⎣ . . . ⎦ = ⎣ . . . ⎦ , (4) ⎣ ... ... ... L (J 1m ⊗ J 1m ) D . . . L (J nm ⊗ J nm ) D vech(kn ) vech(km e ) where L ∈ R6×9 is the the elimination matrix, D ∈ R9×6 is the duplication matrix, and vech(·) is the half-vectorization operator [11]. In the following sections, the system (4) will be treated as a primary one for identification of the desired stiffness parameters. From the first sight, if the number of independent experiments is higher than the number of manipulator links, i.e. m > n, the system (4) should provide the rank 6n that is sufficient for unique estimation of all desired stiffness parameters. Nevertheless, as follows from the motivation example and other relevant studies, the system (4) is always rank-deficient and only limited subset of the desired physical parameters can be identified from the considered experimental data. The latter motivates several theoretical and practical problems that are considered in this paper and deal with (i) identifiability analysis of the link stiffness parameters under different assumptions concerning the ki -matrices, (ii) stiffness model reduction while preserving its completeness property, and (iii) development of some practical techniques for robotic engineers that allows to eliminate the model parametric redundancy and avoid physically unacceptable parameter estimates.

3

Model with 2 × 2 Matrices

Let us consider first the case where the manipulator compliance is described by the model



i i

2  kxy Ci −Si kxx Ci Si cxx cxy = (5) i i cyx cyy Si Ci kxy kyy −Si Ci i=1

where the link/joint stiffness matrices have the size 2 × 2 and the vector of 3n unknown parameters 

1 2 1 1 2 n T kxy kyy kxx kxy . . . kyy . (6) π = kxx In this case, the 3m × 3n regression matrix can be presented in a blockwise form as ⎤ ⎤ ⎡ 2 ⎡ 2 Cij −2Cij Sij Sij A11 . . . A1n i = 1, . . . , n 2 2 − Sij −Cij Sij ⎦ , (7) A = ⎣ . . . . . . . . . ⎦ , Aji = ⎣Cij Sij Cij j = 1, . . . , m 2 2 Am1 . . . Amn Sij 2Cij Sij Cij where Cij and Sij denote the functions cos(·) and sin(·) using the joint variables (q1 , . . . , qn ) at the j-th experiment  i  i     Cij = cos qkj , Sij = sin qkj . (8) k=1

k=1

480

S. Mamedov et al.

As follows from relevant analysis, even if the number experiments m ≥ n is high enough and the manipulator configurations are independent, the regression matrix is rank deficient (rank(A) = 2n + 1 < 3n) and does not allow to identify all unknown coefficients contained in the vector π ∈ R3n . This rank-deficiency is by (n −1) linear relations between the columns of the matrix A =

1caused axx , a1yy , . . . , anyy that may be written as

T a1xx + a1yy = a2xx + a2yy = · · · = anxx + anyy = 1 0 1 .

(9)

Consequently, any partial solution of the considered linear system Aπ = b can produce infinite number of equivalent solutions of the following structure i

 n i kxx + Δi kxy ˆ ki = Δi = 0, i = 1, . . . , n. (10) , i i kxy kyy + Δ i=1

Using these results, it is possible to generate a subset of identifiable parameters π 0 ⊂ π that is in agreement with the rank of the matrix A. As follows from the above expressions, such subset must include: (i) all non-diagonal elements of the matrices ki ; (ii) both diagonal elements of a single arbitrary selected matrix; (iii) one of two diagonal elements from the remaining matrices. It can be proved that the number of possible parameter combinations is equal to N = 2n−1 n.

4

Model with 3 × 3 Matrices

Let us consider now a more general stiffness model, which describes the planar manipulator reaction to both the external force and torque. In the frame of such model, the link/joint compliances are described by the 3 × 3 matrices of the following structure ⎡ i i i ⎤ kxx kxy kxφ i i i ⎦ kyy kyφ , i = 1, . . . , n, (11) ki = ⎣kxy i i i kxφ kyφ kφφ and the vector of unknown parameters π ∈ R6n is 

1 1 1 1 1 1 2 n T kxy kxφ kyy kyφ kφφ kxx . . . kφφ . π = kxx In this case, the link Jacobians may be written as follows ⎡ ⎤ Cij −Sij −yij J ij = ⎣ Sij Cij xij ⎦ , i = 1, . . . , n, j = 1, . . . , m, 0 0 1

(12)

(13)

where i and j denote the link number and the experiment number respectively, and the matrix elements are (8) and xij =

n  k=i+1

Lkj Ckj , yij =

n  k=i+1

Lkj Skj

Stiffness Modeling of Planar Robotic Manipulators

481

are expressed via the joint variables measured at the j-th experiment qij and the link lengths L1 , . . . , Ln . These Jacobians produce the following sub-matrices Aji ⎤ 2 2 Cij −2Cij Sij −2Cij yij Sij 2Sij yij (yij )2 2 2 ⎢Cij Sij Cij − Sij −Sij yij + Cij xij −Cij Sij −Cij yij − Sij xij −xij yij ⎥ ⎥ ⎢ ⎢ 0 0 Ci 0 −Sij −yij ⎥ ⎥ ⎢ Aji = ⎢ 2 2 2Sij xij Cij 2Cij xij (xij )2 ⎥ ⎥ ⎢ Sij 2Cij Sij ⎣ 0 0 Sij 0 Cij −xij ⎦ 0 0 0 0 0 1 ⎡

and the aggregated regression matrix A = [Aji ] has the size 6m × 6n where m > n. As follows from relevant analysis, even for excessive number experiments m  n, the regression matrix is rank deficient (rank(A) = 4n+2 < 6n) and does not allow to identify all unknown coefficients contained in the vector π ∈ R6n . Dedicated analysis shows that this rank-deficiency is caused by 

two types of (n− 1) linear relations between the columns of the matrix A = a1xx a1xy . . . anφφ . The first type of relations is similar to those found for model with 2 × 2 matrices

T a1xx + a1yy = a2xx + a2yy = · · · = anxx + anyy = 1 0 0 1 0 0 , produces equivalent solutions of the following structure ⎡ i i i ⎤ kxx + Δi kxy kxφ n  i i i ⎦ ˆ i = ⎣ kxy kyy + Δi kyφ k , i = 1, . . . , n, Δi = 0. i i i i=1 kxφ kyφ kφφ

(14)

(15)

In contrast, the second type of linear relations i i 2 i ai−1 φφ = aφφ + Li ayφ + Li ayy , i = 2, . . . , n,

produces the equivalent solutions of slightly different form ⎤ ⎡ i i i kxx kxy kxφ i i i ˆ i = ⎣kxy kyy + δi /L2i kyφ + δi /Li ⎦ , {δi ∈ R, i = 2, . . . , n}. k i i i kxφ kyφ + δi /Li kφφ + δi − δi+1

(16)

(17)

Using these results, it is possible to generate a subset of identifiable parameters π 0 ⊂ π that is in agreement with the rank of the matrix A. As follows from the above expressions, such subset may include: (i) all non-diagonal elements i i i i i i , kxφ , kyφ for all matrices ki ; (ii) all diagonal elements kφφ , kyy , kxx of a kxy i i single arbitrary selected matrix; (iii) one of two diagonal elements kxx , kyy from the remaining matrices. It can be proved that the number of possible parameter combinations corresponding to this selection is equal to N = 2n−1 n.

482

5

S. Mamedov et al.

Estimation of Model Parameters

The model reduction techniques presented in the previous sections allow us to achieve agreement between the regression matrix rank and a subset of identifiable parameters. The open question, however, is how to treat the remaining non-identifiable parameters that are obviously redundant mathematically but are present in original model derived from physics. Generally, it is possible to apply one of two possible methods here: (M1) eliminating redundant parameters, which is identical to setting all redundant parameters to zero; (M2) setting the redundant parameters to their nominal values obtained from an approximate physical model. It is clear that both approaches yield mathematically equivalent stiffness models possessing similar input-output relations, but corresponding sets of the parameters may differ essentially. Another possible technique that may be applied here, (M3), is based on minimization of Euclidean norm of relative deviations of the parameter estimates with respect to their nominal values π nom . In this case, the vector of parameters is presented as the product



π = (I + diag(η)) · π nom

(18)

where η = η1 η2 . . . η12 are treated as unknowns and the original linear system Aπ = b is transformed into the rank-deficient system (A · diag(π nom ))η = b − Aπ nom that may be solved via the Moore-Penrose pseudo-inverse ensuring ˆi. ||η|| → min, which intuitively should produce reasonable estimates k To compare these three approaches, let us consider a simple example dealing with a two-link manipulator and 2 × 2 diagonal matrices of parameters. Relevant numerical results are presented in Table 1 and clearly show that a straightforward application of the pseudo-inverse may produce results, which are physically Table 1. Stiffness parameters of a two-link manipulator and their estimates Approach

First link   10 0 Real parameters k1 = 0 2   9 0 nom Nominal parameters k1 = 0 1.5   12.25 0 ˆ Estimates via pseudo-inverse k 1 = 0 4.25   ˆ 1 = 11 0 Setting k 2yy = 0 k 0 3   9 0 2 nom Setting k yy = 3 (nominal) k1 = 0 1.5   9.21 0 ˆ Minimizing norm of ratios k1 = 0 1.21

Second link   20 0 k2 = 0 1   18 0 nom k2 = 0 3   17.75 0 ˆ k2 = 0 −1.25   ˆ 2 = 19 0 k 0 0   18 0 nom k2 = 0 3   20.78 0 ˆ k2 = 0 1.79

Stiffness Modeling of Planar Robotic Manipulators

483

ˆ i is not positive definite). A similar problem unacceptable (one of the matrices k exists for the method M1, which ignores the manipulators link compliances for some arbitrary selected directions. In contrast, the methods M2 and M3 produce physically acceptable results. Hence, in practice, it is reasonable to give preference to either M2 or M3, depending on the a priori knowledge on the model parameters.

6

Application Example

To confirm validity of the developed technique, let us apply it to the elestostatic model calibration for the two-link planar manipulator whose real parameters are presented in Table 2. The data used in estimation was generated in simulation based on real parameters for 10 different manipulator configurations. Before identification, the original complete model with 12 parameters including two symmetrical matrices k1 , k2 was reduced, which yielded the identifiable subset of parameters equal to 8. Further, applying independently the proposed methods M2 and M3, there were obtained the least-square estimates of the matriˆ 2 , which are presented in Table 2. As follows from this table, the proˆ1, k ces k posed technique allowed to resolve the parameter redundancy problem, avoid numerical singularities and obtain the parameter values that are close to the real ones. Table 2. Real and estimated elastostatic parameters of the robot links with E = 7 × 1010 Pa, L1 = 1.075 m, L2 = 1.280 m, S1 = 1.77 × 10−2 m2 , S2 = 0.98 × 10−2 m2 Link 1 compliance 10−6 ⎡ ⎤ 0.0009 0 0 ⎢ ⎥ Real parameters ⎣ 0 0.0523 0.0729 ⎦ 0 0.0729 0.1357 ⎡ ⎤ 0.0027 0 0 ⎢ ⎥ M1 ⎣ 0 0.0541 0.0729 ⎦ 0 0.0729 1.0526 ⎡ ⎤ 0.0007 0 0 ⎢ ⎥ M2 ⎣ 0 0.0521 0.0729 ⎦ 0 0.0729 0.0440 ⎡ ⎤ 0.0009 0 0 ⎢ ⎥ M3 ⎣ 0 0.0523 0.0729 ⎦ 0 0.0729 0.1368 Approach

Link 2 compliance 10−6 ⎡ ⎤ 0.0019 0 0 ⎢ ⎥ ⎣ 0 0.5008 0.5869 ⎦ 0 0.5869 0.9170 ⎤ ⎡ 0.0 0 0 ⎥ ⎢ ⎣ 0 −1.0034 −0.5869 ⎦ 0 −0.5869 0.0 ⎡ ⎤ 0.0021 0 0 ⎢ ⎥ ⎣ 0 0.6512 0.7042 ⎦ 0 0.7042 1.0087 ⎡ ⎤ 0.0018 0 0 ⎢ ⎥ ⎣ 0 0.4988 0.5854 ⎦ 0 0.5854 0.9158

484

7

S. Mamedov et al.

Conclusion

The manipulator elasto-static models, which are widely used in many modern control algorithms, include a number of parameters that are usually not reported by the robot manufacturer and should be estimated from experimental data. However, the conventional identifications techniques based on classical least-square methods cannot be directly applied here because of the parametric redundancy leading to the rank-deficiency of corresponding correlation matrices. This paper deals with the redundancy resolution for such models and proposes new methods of the parameter estimations, which allow being in agreement with physical nature of the considered problem. For two practically important cases dealing with 2 × 2 and 3 × 3 matrices of the link/joint compliances, there were obtained subsets of the mutually dependent parameters and developed techniques that allow to reduce the number of parameters without violating the model completeness property. There were also proposed several methods for parameter estimation that yield physically acceptable results, with positive-definite link/joint compliance matrices. In future, these results will be generalized for spatial serial manipulators, where the elasto-static models include matrices of size 6 × 6 and number of parameters that is essentially larger. Acknowledgement. This work was supported by Russian Foundation for Basic Research (RFBR) grant 18-38-20186.

References 1. Akin, J.E.: Finite Element Analysis with Error Estimators: An Introduction to the FEM and Adaptive Error Analysis for Engineering Students. Elsevier, Burlington (2005) 2. Bouzgarrou, B., Fauroux, J., Gogu, G., Heerah, Y.: Rigidity analysis of T3R1 parallel robot with uncoupled kinematics. In: Proceedings of the 35th International Symposium on Robotics (ISR), Paris, France (2004) 3. Corradini, C., Fauroux, J.C., Krut, S., et al.: Evaluation of a 4-degree of freedom parallel manipulator stiffness. In: Proceedings of the 11th World Congress in Mechanisms and Machine Science, Tianjin, China (2003) 4. Deblaise, D., Hernot, X., Maurine, P.: A systematic analytical method for pkm stiffness matrix calculation. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006, pp. 4213–4219. IEEE (2006) 5. Driels, M.R., Swayze, W.E.: Automated partial pose measurement system for manipulator calibration experiments. IEEE Trans. Robot. Autom. 10(4), 430–440 (1994) 6. Gosselin, C.: Stiffness mapping for parallel manipulators. IEEE Trans. Robot. Autom. 6(3), 377–382 (1990) 7. Gosselin, C.: Stiffness analysis of parallel mechanisms using a lumped model. Int. J. Robot. Autom. 17, 17–27 (2002) 8. Khalil, W., Dombre, E.: Modeling, Identification and Control of Robots. Butterworth-Heinemann, Oxford (2004)

Stiffness Modeling of Planar Robotic Manipulators

485

9. Klimchik, A., Pashkevich, A., Chablat, D.: Cad-based approach for identification of elasto-static parameters of robotic manipulators. Finite Elem. Anal. Des. 75, 19–30 (2013) 10. Lightcap, C., Hamner, S., Schmitz, T., Banks, S.: Improved positioning accuracy of the pa10-6ce robot with geometric and flexibility calibration. IEEE Trans. Rob. 24(2), 452–456 (2008) 11. Magnus, J.R., Neudecker, H.: The elimination matrix: some lemmas and applications. SIAM J. Algebraic Discrete Methods 1(4), 422–449 (1980) 12. Majou, F., Gosselin, C., Wenger, P., Chablat, D.: Parametric stiffness analysis of the Orthoglide. Mech. Mach. Theory 42(3), 296–311 (2007) 13. Marie, S., Courteille, E., Maurine, P.: Elasto-geometrical modeling and calibration of robot manipulators: application to machining and forming applications. Mech. Mach. Theory 69, 13–43 (2013) 14. Martin, H.C.: Introduction to Matrix Methods of Structural Analysis (book on matrix methods of structural analysis including truss, beams, coordinate transformations, force and displacement methods, programming, etc.), p. 331. MCGRAWHill Book Co., New York (1966) 15. Meggiolaro, M.A., Dubowsky, S.: An analytical method to eliminate the redundant parameters in robot calibration. In: Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No. 00CH37065), vol. 4, pp. 3609–3615. IEEE (2000) 16. Park, I.W., Lee, B.J., Cho, S.H., Hong, Y.D., Kim, J.H.: Laser-based kinematic calibration of robot manipulator using differential kinematics. IEEE/ASME Trans. Mechatron. 17(6), 1059–1067 (2011) 17. Pashkevich, A., Chablat, D., Wenger, P.: Stiffness analysis of overconstrained parallel manipulators. Mech. Mach. Theory 44(5), 966–982 (2009) 18. Salisbury, J.K.: Active stiffness control of a manipulator in cartesian coordinates. In: 1980 19th IEEE conference on decision and control including the symposium on adaptive processes, pp. 95–100. IEEE (1980) 19. Santolaria, J., Aguilar, J.J., Yag¨ ue, J.A., Pastor, J.: Kinematic parameter estimation technique for calibration and repeatability improvement of articulated arm coordinate measuring machines. Precision Eng. 32(4), 251–268 (2008) 20. Zhang, D.: Kinetostatic analysis and optimization of parallel and hybrid architectures for machine tools (2001)

Forward Kinematic Analysis of a Rotary Hexapod Alexey Fomin1,2(B) , Anton Antonov1 , and Victor Glazunov1 1 Mechanical Engineering Research Institute of the RAS, Moscow 101000, Russia

[email protected] 2 Siberian State Industrial University, Novokuznetsk 654007, Russia

Abstract. This study presents a solution of the forward position analysis for one-degree-of-freedom rotary hexapod. This mechanism includes six kinematic chains which orient an end-effector (platform) through the use of a single actuator. The forward position analysis is partially carried out by analytical and numerical methods using MATLAB package. The obtained solution has been demonstrated on three case-studies when the end-effector has exclusively rotational movement around the vertical axis, simultaneous rotational and linear movements also relative to the vertical axis and a full range of movements relative to each axis. The results of the analytical and numerical calculations have been validated by the findings obtained under the analysis of a virtual prototype of the rotary hexapod. Keywords: Forward kinematics · Position analysis · Degree-of-freedom · Parallel mechanism · Rotary hexapod · MATLAB simulation

1 Introduction One-degree-of-freedom (one-DoF) mechanisms are widely used in practice. Those mechanisms operate with a single drive. Their structural properties allow implementing various types of technological operations that require cyclic movements of end-effectors along prescribed trajectories. In particular, one-DoF mechanisms find wide application in devices where endeffectors are guided along planar trajectories. Crushing machines for mining and metallurgical industries [1–3], metallurgical shears [4], combustion engines [5, 6], mechanized devices for food industry [7, 8], robotic systems of various profiles [9–11], deployable mechanical systems [12–14], agricultural machines [15, 16] and many other technological devices are designed based on these mechanisms. One-DoF mechanisms are also extensively used in machinery having spatial movements of end-effectors. Examples of such machinery include various shaking tables and vibrating screens [17, 18], orienting mechanisms [19], biomedical equipment [20, 21], systems for processing machine elements [22], various technique for aerospace applications [23–25] and a number of other technological systems [26–28]. Obviously, the development and investigation of one-DoF mechanisms is quite essential problem that covers many important technological areas. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 486–494, 2021. https://doi.org/10.1007/978-3-030-58380-4_58

Forward Kinematic Analysis of a Rotary Hexapod

487

In the presented study, a one-DoF rotary hexapod is analyzed, its virtual prototype (computer-aided design (CAD) model), developed in Autodesk Inventor, is shown in Fig. 1. The hexapod includes fixed link 1 made as a circular guide, driving link made as driving wheel 2 and six kinematic chains. Each chain includes gear 3 and drive pulley 4, having common spindle, driven pulley 5 and crank 6, also having common spindle, link block 7, swinging arm 8 and carriage 9, which are rigidly connected, and rod 10. The end effector is platform 11. Pulleys 4 and 5 are connected by flexible coupling. Rod 10 forms spherical joints on both sides with carriage 9 and platform 11. Points K i and E i with i = 1 … 6 determine the centers of spherical joints 9-10 and 10-11. Base coordinate system O1 x 1 y1 z1 is connected to fixed link 1. The local coordinate system OP x P yP zP is connected to platform 11, point P is the center of platform 11. Figure 1 also presents coupling of driving wheel 2 and horizontal part of one kinematic chain (links 3-9) that is placed within circular guide 1.

Fig. 1. Virtual prototype (CAD model) of the one-DoF rotary hexapod (left view); Planar kinematic chain of the hexapod placed within circular guide (right view).

The structure and operating principle of the hexapod are described in [29]. In the hexapod the movement from driving wheel 2 is transmitted to each of the six kinematic chains orienting platform 11 in space. The uniqueness of the hexapod is in the opportunity to get different law motions of the end-effector while having single actuator. This happens because the flexible connection between pulleys 4 and 5 in each kinematic chain is enabled to be unlinked and cranks 6 might be reoriented. Hereby, new law motions of the end-effector could be obtained. The aim of this study is to carry out forward position analysis of the hexapod.

2 Analytical and Numerical Solution of the Forward Position Analysis This section provides forward position analysis, which consists in determining the coordinates of platform 11 for predetermined rotational angle of the motor shaft. First of

488

A. Fomin et al.

all, consider how to set the position and orientation of the end-effector. The position of platform 11 can be described by the Cartesian coordinates of its geometric center, point P, relative to the base coordinate system T  (1) pP = xP yP zP , where pP is the vector determining the position of point P in base coordinate system O1 x 1 y1 z1 ; x P , yP , and zP are coordinates of point P in base coordinate system O1 x 1 y1 z1 . Rotation matrix RP between base coordinate system O1 x 1 y1 z1 and local coordinate system OP x P yP zP connected to platform 11 uniquely determines local axes orientation. There are different approaches of representing the orientation of a solid body. Accordingly, there are various methods that allow forming matrix RP [30]. In this study, the orientation of the hexapod’s end-effector will be set through Euler angles, then matrix RP can be written as the following matrix composition RP = RO1z1 (ϕ)RO1 y1 (θ)RO1 x1 (ψ),

(2)

where O1 y1 is the axis of the rotated coordinate system after rotation of the end-effector by angle ϕ around axis O1 z1 ; O1 x1 is the axis of the rotated coordinate system after rotation of the end-effector by angle θ around axis O1 y1 ; ϕ, θ, and ψ are Euler angles of the end-effector. The coordinates of the end-effector can be written as a single vector T  (3) X = xP yP zP ϕ θ ψ . Introduce parameter q as the rotational angle of the motor shaft. Then the solution of the forward position problem will be in determining end-effector’s coordinates X depending on rotational angle q X = f(q),

(4)

where f(q) is the vector function that determines the dependence of the end-effector’s coordinates on rotational angle q of the motor shaft. Consider features of the forward position analysis of the hexapod. Discuss kinematic analysis of its planar part. Rotational angles γ1i of gear 3 and γ2i of driven pulley 5 (Fig. 1) could be found in the following way γ1i = qR2 /R3i , γ2i = γ1i R4i /R5i ,

(5)

where i = 1…6 is the numerical order of a kinematic chain; R2 is the radius of driving wheel 2; R3i is the radius of gear 3; R4i is the radius of driving pulley 4; R5i is the radius of driven pulley 5. Since center O1 of driving wheel 2, center Ai of driving pulley 4 and center Bi of driven pulley 5 are on one straight line (Fig. 1), the distance O1 Bi between the axes of driving wheel 2 and driven pulley 5 can be determined as O1 Bi = R2 +R3i +d i , where d i is the distance between Ai and Bi (i.e. between the centers of pulleys 4 and 5). Based on the law of cosines for triangle O1 Bi C i the length of O1 C i is defined as  (6) O1 Ci = O1 Bi2 + li2 + 2O1 Bi li cosβi ,

Forward Kinematic Analysis of a Rotary Hexapod

489

where l i is the length of Bi C i ; βi is the rotational angle of crank 6 calculated as follows βi = β0i − γ2i , where β0i is the start angle of rotation of crank 6. Then define angle δi between O1 Bi and O1 C i through the law of cosines for triangle O1 Bi C i (7) Coordinates of point K i of carriage 9 in base coordinate system O1 x 1 y1 z1 could be written as T  (8) pKi = R1 cos(αi + δi ) R1 sin(αi + δi ) 0 , where R1 is the radius of circular guide 1; αi is the angle between axis O1 x 1 and O1 Bi . Coordinates of point E i of platform’s joint 10-11 in base coordinate system O1 x 1 y1 z1 could be expressed as pEi = pP + RP rEi ,

(9)

where i = 1…6 is the numerical order of a kinematic chain; rEi are the coordinates of point E i in local coordinate system OP x P yP zP . Finally, it is possible to compose a relation linking the coordinates in two expressions shown above with rod length L i (pEi − pKi )2 = L2i .

(10)

This expression, written for each of the kinematic chains, is a system of six equations, where coordinates of carriages pKi depend only on rotational angle q of the motor shaft, and coordinates pEi depend on platform’s coordinates X. In solving the forward position problem, it is required to express the components of vector X from the above relations. However, obtaining an analytical solution is associated with certain difficulties [31], and therefore numerical methods are used. In this case the last equation can be presented in the following view  2 fi (X, q) = pEi (X) − pKi (q) − L2i , i = 1 . . . 6. (11) The numerical solution of the forward position problem is in searching a solution of the following equation system fi (X, q) = 0, i = 1 . . . 6.

(12)

Various numerical methods are known (Newton’s method, least squares method, interval analysis methods, etc.) and examples of their application [31–34]. It is necessary to set the initial approximation, vector X0 , to determine the numerical solution. It should be noted that the choice of this approximation can significantly affect the solution of the compiled system of equations. For the solution of the forward position problem it indicates the possibility of different assemblies of the hexapod with the same value of rotational angle q of the motor shaft. The task of determining all possible solutions to this problem (all possible assemblies of the hexapod) is not included in this study, however some examples can be found in [35, 36].

490

A. Fomin et al.

3 Case-Study: Forward Position Analysis Depending on Different Orientation of the Base Links Consider several solution examples of the forward position problem for various initial positions of cranks 6, i.e. for different angles β0i . Other geometric parameters correspond to the CAD model of the hexapod according to Fig. 1 and are shown in Table 1. In all calculations, one revolution of the motor shaft with a constant speed of 3 rpm as a law motion has been set. The solution has been carried out in MATLAB package using standard function «fsolve». The initial approximation at zero step of the calculation has been chosen as follows  T (13) X0 = 0 0 180 0 0 0 . In this case, the result of the numerical solution for the current value of the rotational angle of the motor shaft has been used as an initial approximation for the next value of this angle. Table 1. Geometrical parameters of the rotary hexapod. i

1

2

R1 , mm

246.00

R2 , mm

64.25

R3i , mm

24.00

R4i , mm

30.00

R5i , mm

15.00

d i , mm

75.50

l i , mm

39.00

αi , deg rEi , mm

30 90 T  R11 cosχi R11 sinχi 0

R11 , mm

192.70

χi , deg

10

L i , mm

220.00

110

3

4

5

6

150

210

270

330

130

230

250

350

Notice: R11 is the radius of platform 11; χi is the angular placement of the spherical joints on platform 11.

In the first case, the angles β0i are set to equal each other, particularly β0i = 255.8º, i = 1…6. Obtained calculation data is diagrammatically presented in Fig. 2. It comes from Fig. 2 that platform 11 performs only rotational oscillatory motion relative to axis O1 z1 . Moreover, the number of oscillations at the specified interval of the calculation exceeds the number of revolutions of the motor shaft. This happens due to the ratio of the diameters of the wheels and pulleys.

Forward Kinematic Analysis of a Rotary Hexapod

491

Fig. 2. Diagram of coordinates variation of the end-effector when it realizes only rotational movement around the vertical axis.

The following values of the start angles are accepted for the second calculation: β01 = 0 β3 = β05 = 104.2º and β02 = β04 = β06 = 255.8º. In this case the end-effector performs linear and rotational oscillatory movements relative to axis O1 z1 . Figure 3 shows the diagram describing this case.

Fig. 3. Diagram of coordinates variation of the end-effector when it realizes only rotational and linear movements relative to the vertical axis.

For the third calculation, the start angles have been randomly chosen: β01 = 287.6º, β02 = 103.3º, β03 = 278.4º, β04 = 263.8º, β05 = 260.6º, β06 = 242.4º. In accordance with these values, the diagram of coordinate variation of the end-effector (platform 11) has been obtained. It is shown in Fig. 4. It comes from this diagram that the position and tilt angles of the endeffector change relative to each axis.

492

A. Fomin et al.

Fig. 4. Diagram of coordinates variation of the end-effector when it realizes rotational and linear movements relative to each coordinate axis.

Hereby, it is possible to achieve different trajectories of the end-effector in sixdimensional space by varying the initial position of cranks 6. The results of analytical and numerical calculations of the forward position problem for the three considered cases have been completely validated by CAD modelling. For this purpose the virtual prototype shown in Fig. 1 has been used.

4 Conclusions The presented study provides a solution of the forward position problem for the rotary hexapod having single drive. The problem has been partially solved by analytical and numerical methods using MATLAB package. In accordance with the obtained solution, three cases in which the end-effector has different types of movements in six-dimensional space have been considered. These cases include only rotational movement of the endeffector around the vertical axis, simultaneous rotational and linear movements also relative to the vertical axis, and a full range of movements relative to each axis. Diagrams of coordinate variation of the end-effector have been obtained for each case. The results of analytical and numerical calculations have been proved by the computational data of the virtual prototype (CAD model) of the hexapod. Future research could be associated with task solution of velocities and accelerations, as well as with further dynamic analysis of the hexapod. Acknowledgements. The study has been carried out with the support of the Russian President Grant according to the research project MK-2781.2019.8.

References 1. Refahia, A., Aghazadeh Mohandesi, J., Rezai, B.: Discrete element modeling for predicting breakage behavior and fracture energy of a single particle in a jaw crusher. Int. J. Miner. Process. 94(1–2), 83–91 (2010)

Forward Kinematic Analysis of a Rotary Hexapod

493

2. DeDiemar, R.B.: New concepts in Jaw Crusher technology. Miner. Eng. 3(1–2), 67–74 (1990) 3. Johansson, M., Bengtsson, M., Evertsson, M., Hulthen, E.: A fundamental model of an industrial-scale jaw crusher. Miner. Eng. 105, 69–78 (2017) 4. Kuznetsov, A.P., Kosov, M.G.: Structural precision of metal-cutting machines. Russ. Eng. Res. 32(5–6), 482–490 (2012) 5. Hoeltgebaum, T., Simoni, R., Martins, D.: Reconfigurability of engines: a kinematic approach to variable compression ratio engines. Mech. Mach. Theor. 96(2), 308–322 (2016) 6. Ma, Z.D., Perkins, N.C.: An efficient multibody dynamics model for internal combustion engine systems. Multibody Syst. Dyn. 10(4), 363–391 (2003) 7. Fomin, A., Ivanov, W.: Development of a mixing mechanism with a complex motion of the end-effector. Strojniski Vestnik/J. Mech. Eng. 65(5), 319–325 (2019) 8. Gao, F., Liu, X., Gruver, W.A.: Performance evaluation of two-degree-of-freedom planar parallel robots. Mech. Mach. Theor. 33(6), 661–668 (1998) 9. Staicu, S.: Inverse dynamics of the 3-PRR planar parallel robot. Robot. Auton. Syst. 57(5), 556–563 (2009) 10. Kotlarski, J., Abdellatif, H., Ortmaier, T., Heimann, B.: Enlarging the useable workspace of planar parallel robots using mechanisms of variable geometry. In: ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots (2009) 11. Tatlicioglu, E., Walker, I.D., Dawson, D.M.: Dynamic modelling for planar extensible continuum robot manipulators. In: IEEE International Conference on Robotics and Automation (2007) 12. St-Onge, D., Gosselin, C.: Synthesis and design of a one degree-of-freedom planar deployable mechanism with a large expansion ratio. J. Mech. Robot. 8(2), 021025 (2016) 13. Li, J., Huang, H., Yan, S., Yang, Y.: Kinematic accuracy and dynamic performance of a simple planar space deployable mechanism with joint clearance considering parameter uncertainty. Acta Astronaut. 136, 34–45 (2017) 14. Garcia-Mora, C.J., Sanchez-Sanchez, J.: Geometric method to design bistable and non bistable deployable structures of straight scissors based on the convergence surface. Mech. Mach. Theor. 146, 103720 (2020) 15. Celik, A.: Design and operating characteristics of a push type cutter bar mower. Can. Biosyst. Eng. 48, 23–28 (2006) 16. Guarnieri, A., Maglioni, C., Molari, G.: Dynamic analysis of reciprocating single-blade cutter bars. Trans. ASABE 50(3), 755–764 (2007) 17. Glazunov, V., Kraynev, A.: Design and singularity criteria of parallel manipulators. In: Zielinska, T., Zielinski, C. (eds.) Romansy 16. CISM Courses and Lectures, vol. 487. pp. 15–22. Springer, Vienna (2006) 18. Zahedia, S.A., Babitsky, V.: Modeling of autoresonant control of a parametrically excited screen machine. J. Sound Vib. 380, 78–89 (2016) 19. Fomin, A., Petelin, D., Glazunov, V.: Calculation of drive mechanism for novel rotary hexapod with single motor. In: Zawislak, S., Rysinski, J. (eds.) Engineer of the XXI Century. Mechanisms and Machine Science, vol. 70, pp. 17–28. Springer, Cham (2020) 20. Sancisi, N., Zannoli, D., Parenti-Castelli, V., Belvedere, C., Leardini, A.: A one-degree-offreedom spherical mechanism for human knee joint modelling. Proc. Inst. Mech. Eng. Part H J. Eng. Med. 225(8), 725–735 (2011) 21. Wolbrecht, E.T., Reinkensmeyer, D.J., Perez-Gracia, A.: Single degree-of-freedom exoskeleton mechanism design for finger rehabilitation. In: IEEE International Conference on Rehabilitation Robotics (ICORR), 5975427 (2011) 22. Fomin, A.S., Paramonov, M.E.: Structural and kinematic analysis of a mechanism for internal surfaces processing. J. Mach. Manuf. Reliab. 48, 292–298 (2019)

494

A. Fomin et al.

23. Zirbel, S.A., Trease, B.P., Magleby, S.P., Howell, L.L.: Deployment methods for an origamiinspired rigid-foldable array. In: Proceedings of the 42nd Aerospace Mechanism Symposium, pp. 189–194 (2014) 24. Kiper, G., Soylemez, E.: Deployable space structures. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies, pp. 131–138 (2009) 25. Li, T.: Deployment analysis and control of deployable space antenna. Aerosp. Sci. Technol. 18(1), 42–47 (2012) 26. Marras, A.E., Zhou, L., Su, H.-J., Castro, C.E.: Programmable motion of DNA origami mechanisms. Proc. Natl. Acad. Sci. 112(3), 713–718 (2015) 27. Tounsi, M., Hbaieb, R., Chaari, F., Fakhfakh, T., Haddar, M.: Dynamic stability analysis of a flexible cam mechanism using a one-degree-of-freedom model. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 223(5), 1057–1068 (2009) 28. Zarrouk, D., Shoham, M.: Analysis and design of one degree of freedom worm robots for locomotion on rigid and compliant terrain. J. Mech. Des. 134(2), 021010 (2012) 29. Fomin, A., Glazunov, V., Terekhova, A.: Development of a novel rotary hexapod with single drive. In: Arakelian, V., Wenger, P. (eds.) ROMANSY 22 – Robot Design, Dynamics and Control. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 584, pp. 141–146 (2018) 30. Waldron, K.J., Schmiedeler, J.: Kinematics. In: Siciliano, B., Khatib, O. (eds.) Springer Handbook of Robotics. Springer Handbooks, pp. 11–36. Springer, Cham (2016) 31. Merlet, J.-P.: Parallel Robots, 2nd edn. Springer, Dordrecht (2006) 32. Merlet, J.-P.: Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int. J. Robot. Res. 23(3), 221–236 (2004) 33. Rolland, L., Chandra, R.: On solving the forward kinematics of the 6-6 general parallel manipulator with an efficient evolutionary algorithm. In: Parenti-Castelli, V., Schiehlen, W. (eds.) ROMANSY 18 Robot Design, Dynamics and Control, pp. 117–124. Springer, Vienna (2010) 34. Zhu, Q., Zhang, Z.: An efficient numerical method for forward kinematics of parallel robots. IEEE Access 7, 128758–128766 (2019) 35. Dietmaier, P.: The Stewart-Gough platform of general geometry can have 40 real postures. In: Lenarcic, J., Husty, M.L. (eds.) Advances in Robot Kinematics: Analysis and Control, pp. 7–16. Springer, Dordrecht (1998) 36. Merlet, J.-P.: Direct kinematics and assembly modes of parallel manipulators. Int. J. Robot. Res. 11(2), 150–162 (1992)

Kinematics of 2-DOF AGVs with Differential Driving Wheels and Caster Wheels Modeling Mohammadreza Montazerijouybari1(B) , Luc Baron1 , and Sousso Kelouwani2 1

2

Polytechnique Montr´eal, PO box 6079, Station CV, Montr´eal, QC H3C 3A7, Canada {mohammadreza.montazerijouybari,Luc.Baron}@polymtl.ca Universit´e du Qu´ebec ` a Trois-Rivi`eres, Trois-Rivi`eres, QC G8Z 4M3, Canada [email protected]

Abstract. Automated Guided Vehicles (AGV) are increasingly used in the industry for a wide variety of applications such as goods handling and manufacturing. Under the assumption of no slipping and no skidding, the kinematic model does not involve any geometric parameter of caster wheels. However, when the vehicle is carrying heavy loads not well distributed on it, the interaction between caster and driving wheels become significant, and needed to be taken into account. As a first step, this paper proposes a mathematical model based on the kinematics in order to predict the behavior of caster wheels. Simulation results show good prediction of caster wheel orientations in some scenarios relative to the dynamic model of the Webots simulator.

Keywords: Caster wheel

1

· Differential wheel · Kinematic modeling

Introduction

Automated guided vehicles (AGV) are increasingly used in the industry for a wide variety of applications such as goods handling and manufacturing. Several configurations exist for incorporating the motorized driving wheels together with passive wheels, either fixed on the vehicle or free to rotate as implemented by caster wheels. The latter is often included as a passive mean for the stability of the load carried and its inheritance of not affecting the control law, under the assumption of no slipping and no skidding. However, when carrying heavy loads not well centered on the vehicle, the interaction between the caster wheels and the driving wheels becomes significant, and needed to be taken into account. Since a caster wheel is freely rotating, the necessity to estimate its orientation would allow a better knowledge of the caster wheel-ground interactions. Kinematic and dynamic behaviors of differential wheels integrated with caster wheels were well studied in the literature [1–5]. All of them are getting ride of the caster wheel behavior, and hence, are not able to consider this significant interaction c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 495–502, 2021. https://doi.org/10.1007/978-3-030-58380-4_59

496

M. Montazerijouybari et al.

with the driving wheels when carrying heavy loads [6]. In this paper, we concentrated on the modeling of the kinematic behavior of caster wheels in order to estimate their orientation based only on kinematic means. A long term objective is to incorporate such a mathematical model into control schema of the AGV in order to carry heavy loads not necessarily well centered on the platform. As shown in Fig. 1, the left and right sides of the platform are symmetric, while the front and rear sides are different. The platform is controlled by two motorized differential driving wheels that are aligned along a common revolute axis. The platform has four passive caster wheels spread out over the four corners of the platform in order to improve the load carrying stability. caster pivots

differential driving wheels

B @ B@ B @ BN @ @ left @ @ @ @ @R

caster pivots

A@ A@ A @ A @ AU @

@ @

@ @R

rear

A@ A@ AU @ @

@I @ @ @

front

@

@ @ @R

@I @ @ @ right

@ @ @

BM B

B @ B @B @B

passive caster wheels

@ AK @A @ A

passive caster wheels

Fig. 1. Automatic Guided Vehicule (AGV) with two motorized differential driving wheels and four passive caster wheels.

2

Problem Formulation

As depicted in Fig.2, let point Pi designates the pivot point of the caster wheel i, point Dj the contact point of the driving wheel j with the ground, while points OA and OB , are respectively, the origins of frames A fixed to the ground and frame B attached to the platform. The two independently actuated differential driving wheels are located along the y-axis of B.

Kinematic Modeling of AGVs with Differential and Caster Wheels

497

D2

e

left P4

P2 c

d

j

B

OB

rear

i

P3 y OA

right b

A

front

P1

a D1

x

Fig. 2. Geometry of the automatic guided vehicule (AGV) under study.

3

Kinematic Analysis

Under the assumption of no slipping and no skidding of the wheels, we have a 2Degree-Of-Freedom (DOF) mobile platform, although it can reach any position and orientation in the plane, i.e. three unknowns of position and orientation. This under actuation is the fundamental characteristic of a nonholonomic mechanical systems such as wheeled mobile platforms. Since the kinematic formulation is easier expressed in the moving frame B, let us define i, j and k as unit vectors along x- , y- and z-axis of frame B, i.e., i = [cos ψ sin ψ 0]T ,

j = [− sin ψ cos ψ 0]T ,

k = [0 0 1]T ,

(1)

where ψ is the orientation of the platform. For a planar motion, k is identical in frames A and B. The radius of the driving wheels is rd , while the one of caster wheels is rc . 3.1

AGV Modeling

We follow the kinematic modeling of [7,8] that we summarize below for quick reference. The linear velocity v of OB relative to of OA and the angular velocity ψ˙ of the platform can be written in moving frame B as rd ˙ (θ1 + θ˙2 )i 2

(2)

rd ψ˙ = (θ˙1 − θ˙2 )k 2c

(3)

v=

498

M. Montazerijouybari et al.

with θ˙1 and θ˙2 are the actuated revolute speed of the driving wheels. It is possible to write the motion of the platform as a relationship between the twist t, the actuated driving wheel speeds θ˙ and the Jacobian matrix J as t ≡ [ψ˙

T

˙ vT ]T = Jθ,

θ˙ ≡ [θ˙1 θ˙2 ]T

where t and θ˙ are, respectively, 6- and 2-dimensional columns, while J is   rd k −k J= 2c ci ci 6×2

(4)

(5)

Since the motion is planar, the first two rows and the last row of J are useless, and hence, a reduced formulation is sufficient. Equation (4) becomes ˙ tP = JP θ, where tP and JP are a 3-dimensional column and a 3 × 2 matrix, i.e., ⎡ ⎤ 1 −1 r d ⎣ c cos ψ c cos ψ ⎦ ˙ T , JP = tP ≡ [ψ˙ x˙ y] 2c c sin ψ c sin ψ 3×2

(6)

(7)

Apparently, the kinematic model of the AGV does not depend on caster wheels, neither their locations, nor their orientations. Consequently, a kinematic model is sufficient for navigation control for most regular operations. However, for heavy-load operations, a dynamic model that include caster wheel kinematics is required to calculate the inertia matrix, the centrifugal and Coriolis forces acting on the platform. Unfortunately, the caster wheel orientation (also called steering angle) is unknown. In a recent work [9], a computational scheme is proposed to estimate the steering angle along continuous paths. An AGV with differential driving wheels is able to slow down and aligned its caster wheels along the velocity of the pivot points, and then restart toward another direction with another orientation of the velocity of these pivots. In this situation, the steering angle change drastically. In the next section, we propose a computational scheme to estimate the caster wheel orientation. 3.2

Caster Wheel Modeling

As shown in Fig. 3, the AGV is turning left around the instantaneous center of rotation (ICR) point O. The distance to each caster pivot is ri , while r is the distance to point OB . The orientation of each caster wheel is given by ψi . Start Turing : When the vehicle start turning, the velocity of pivot Pi , with respect to the point O, is the result of a pure rotation such as   sin(αi − ψ) ˙ vPi = ri ψ (8) cos(αi − ψ)

Kinematic Modeling of AGVs with Differential and Caster Wheels

499

ICR ψ2

r2

O

P2

α2 r4

r1 i

α4

B

ψ4

r3

ψ

j α1

ψ1 P1

OB

P4

r

front

α3 rear

ψ3 P3 y OA

A x

Fig. 3. Caster wheels orientations relative to the instantaneous center of rotation (ICR).

On the other hand, the velocity of each caster pivot could be decomposed into two orthogonal components. First, rc θ˙ which is the result of caster wheel heading velocity. Second, eψ˙ i which is the consequence of the rotation of the caster wheel around the contact point to the ground. Expressing these two velocities in frame A yields   r θ˙ cos ψi − eψ˙ i sin ψi (9) vPi = c ˙i rc θi sin ψi + eψ˙ i cos ψi Once comparing Eqs. (8) and (9), we obtain tan ψi =

ri ψ˙ cos(αi − ψ) − eψ˙ i cos ψi ri sin(αi − ψ) + eψ˙ i sin ψi

(10)

When the AGV start turning, ψi rises and reaches its final orientation [ψi ]f and ψ˙ i get back to zero. Applying this condition to the Eq. (10) we have [ψi ]f =

π + ψ − αi , 2

(11)

500

M. Montazerijouybari et al.

with r+d r−d ), α2 = tan−1 ( ), (12) a a r+d r−d ), α4 = tan−1 ( ), (13) α3 = tan−1 ( b b The lateral displacement of Pi , in A, from a rotation around ICR, has to be equal to the displacement of the same point from the caster wheel rotation around the wheel ground contact point, i.e., ri (sin(αi − ψ) − sin αi ), (14) sin ψi − sin [ψi ]init = e α1 = tan−1 (

where [ψi ]init is the initial value of ψi . Start Going Straight: When the vehicle start going straight forward after turning, the caster wheels are not yet in neutral orientation, but will do so with a kinematic relationship. As shown in Fig. 3, each caster wheel pivot Pi must follow the vehicle, while the heading velocity of the wheel center point is given by rc θ˙i and the velocity of the caster wheel caused by rotation around the contact point to the ground is eψ˙ i . Since the sum of these two must results in the velocity rd θ˙ of the vehicle as Along i : Along j :

rd θ˙ = rc θ˙i cos ψi + eψ˙ i sin ψi eψ˙ i cos ψi = rc θ˙i sin ψi

(15) (16)

Upon substitution of Eq. (16) into (15) in order to eliminate θ˙i , we obtain rd θ˙ ψ˙ i = e sin ψi

(17)

The integral relative to time of Eq. (17) between the initial condition, which we assume [ψi ]f from previous stage, and the final condition ψi , we have  ψi  ψ˙ i rd θ˙ = (18) e [ψi ]f sin ψi rd θ = ln[tan([ψi ]f /2)] − ln[tan(ψi /2)] e

4

(19)

Simulation Results

The computational scheme of Eqs. (11), (14) and (19), for estimating the caster wheel orientation ψi is compared with the two different scenarios from a dynamic simulation implemented on the Webot platform [10]. The later is a free opensource 3D robot simulator platform initially developed by EPFL in 1996. It uses an open dynamic engine to accurately simulate physical properties of objects and multi-body dynamics of robotic systems. The geometry of the AGV is given as: a = 631 mm, b = 562 mm, c = 344 mm, d = 250 mm, e = 34 mm, rc = 38 mm and rd = 100 mm.

Kinematic Modeling of AGVs with Differential and Caster Wheels

501

Scenario 1: The platform is going straight and start turning with two different radius. For an easy comparison, we locate a) the ICR at point OB so the AGV is turning around itself (r = 0 mm); b) the ICR at point Dj so that the platform turns around one of its driving wheels contact point to the ground (r = 344 mm). Scenario 2: The platform starts going straight forward after turning with two different radius c) from r = 0 mm; and d) from r = 344 mm. ψi

ψi

(a) senario 1 : r = 0 mm; ψi

(b) senario 1 : r = 344 mm; ψi

(c) senario 2 : r = 0 mm;

(d) senario 2 : r = 344 mm;

Fig. 4. Mathematical model and Webot’s dynamic simulation : scenario 1 from the neutral orientation of the caster wheels a) turning with r = 0 mm; b) turning with r = 344 mm; scenario 2 going straight c) after turning with r = 0 mm; and d) after turning with r = 344 mm. All angles are expressed in degrees.

Apparently from Fig. 4, the mathematical model is able to estimate the caster wheel orientation based only on kinematic relationships as compare with the results from a Webot multi-body dynamic simulation. In scenario 1, the caster wheel start from the neutral position. For the two radius, the caster wheel orientation quickly rotate to satisfy the geometric compatibility of the ICR. After, they remain constant while rotating the AGV. In scenario 2, the caster

502

M. Montazerijouybari et al.

wheel start from a geometric compatibility of the ICR, i.e., ψi = 0. These values depends on the initial radius r. Again here, the caster wheel angles quickly go to zero while the AGV is going straight forward. It is noteworthy that the simulation rate of the mathematical model and the dynamic simulation platform are not the same, thus explaining time differences.

5

Conclusions

We have presented a new mathematical model base on kinematics in order to predict the behavior of caster wheels, and more specifically its orientation for two scenarios. First, start turning by a known radius and second, going straight forward after turning. The mathematical model is able to capture the main behavior of the motion, but still other approaches remains to explore in order to have a simple prediction of the reaction of caster wheels. The validity of mathematical model has been verified by running the full model of AGV in simulation environment using Webots.

References 1. Gentile, A., Messina, A., Trentadue, A.: Dynamic behavior of a mobile robot vehicle with a two caster and two driving wheel configuration. Veh. Syst. Dyn. 25(2), 89– 112 (1996). https://doi.org/10.1080/00423119608968959 2. Chen, P.-C., Huang, H.-P.: T3D dynamical analysis for a caster wheeled mobile robot moving on the frictional surface. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, October 2006, Beijing, China (2006). https://doi. org/10.1109/IROS.2006.282244 3. Angeles, J.: Fundamentals of Robotic Mechanical Systems : Theory, Methods, and Algorithms, 2nd edn. Springer, New York (2003) 4. Kulkarni, A.V.: Instant center based kinematic and dynamic motion synthesis for planar mobile platforms. Ph.D. thesis, University of Texas (2009) 5. Zaw, M.T.: Kinematic and dynamic analysis of mobile robot. M.Sc. thesis, National University of Singapore (2003) 6. Naraghi, M.: Dynamics and control of fast automated guided vehicles for high load applications. Ph.D. thesis, University of Ottawa (1996) 7. Saha, S., Angeles, J.: Kinematics and dynamics of a three-wheeled 2-DOF AGV. In: IEEE Proceedings, International Conference on Robotics and Automation, Scottsdale, AZ (1989). https://doi.org/10.1109/ROBOT.1989.100202 8. Saha, S., Angeles, J., Darcovich, J.: The kinematic design of a 3-DOF isotropic mobile robot. In: Proceedings IEEE International Conference on Robotics and Automation (1993). https://doi.org/10.1109/ROBOT.1993.291996 9. Wu, X., Angeles, J., Zou, T., Xiao, H., Li, W., Lou, P.: Steering-angle computation for the multibody modelling of differential-driving mobile robots with a caster. Int. J. Adv. Robot. Syst. (2018). https://doi.org/10.1177/1729881418820166 10. Cyberbotics Ltd : WEBOT an open source and multi-plaform desktop application used to simulate robots, A spin-off of EPFL (Ecole Polytechnique Federal de Lauzanne), Switzerland. https://cyberbotics.com/. Accessed Febuary 2020

Motion Synthesis Using Low-Dimensional Feature Space and Its Application to Inverse Optimal Control Soya Shimizu1,2(B) , Ko Ayusawa2 , and Gentiane Venture1,2 1

2

Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, Fuchu, Japan [email protected] CNRS-AIST Joint Robotics Laboratory, UMI3218/RL Intelligent Systems Research Institute, National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

Abstract. Inverse Optimal Control (IOC) is a useful technique to estimate the importance of cost functions used to optimally control a system by observing its motion. The computational cost of IOC increases with the complexity of the system. This study aims at analyzing a complex multiple joint motion and estimating weights of the cost functions easily in a low-dimensional feature space using Functional Principal Component Analysis (FPCA). Each motion data is represented by a point in the FPC space, which enables to synthesize easily various motions. In addition, our method allows to estimate the weights of the cost functions for the subject motion with low computational cost compared to traditional methods for solving the IOC problem. To demonstrate our method we apply it to a humanoid robot arm with considering 3 cost functions, and we evaluate the accuracy and the calculation speed of solving the IOC problem.

Keywords: Inverse Optimal Control

1

· Motion synthesis · FPCA

Introduction

Humanoids have been recently used as human physical dummies for evaluating human assistive devices [1,2]. Such uses require to be able to generate stable coordinated whole body motions which is not always trivial [3,4]. Direct Optimal Control (DOC) is one useful approach which generates motion trajectories with considering stability of the humanoid robot by solving an optimization problem [5]. The objective function of the DOC problem consists of a weighted sum of various costs made of motion quantities like joint angles, velocities, kinetic energy, and so on. While the start and end point remain unchanged, by changing the weights, a new trajectory can be generated. Though it is often said that human movements are regulated by DOC principles, it is difficult to generate human c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 503–511, 2021. https://doi.org/10.1007/978-3-030-58380-4_60

504

S. Shimizu et al.

like motions by randomly choosing weights and costs. It is because humans are suspected to minimize an objective with weights and costs that are changing over time [6,7]. One possible option to generate human-like motions is to retrieve the weights for a set of subject costs directly from human movements. Inverse Optimal Control (IOC) is one useful approach for doing so [8–11]. IOC calculates the weights of the subject costs from observed motion data. It can be evaluated which pair of weight-cost is regarded as important in a human motion by IOC. It is also possible to generate human-like motions by DOC using the weight-costs which are obtained by IOC. However, IOC is computationally costly, so it is not practical to straightforwardly apply IOC to whole body human/humanoid motion. For those reasons, this paper proposes a new method for estimating weights in IOC problem. It is based on the use of Functional Principal Component Analysis (FPCA) instead of directly using IOC. FPCA is one of the data analysis methods for functional data, high dimensional smoothed data represented by a functions, i.e. B-spline [12–15]. FPCA converts each functional data into a point in a low dimensional space called FPC space. This point includes the characteristics of the original data. The method was used to synthesize whole body humanoid motion motion data by mixing different points in the FPC space [16]. Furthermore, motion synthesis considering physical consistency such as whole-body balancing conditions could be achieved by FPCA [17]. The proposed framework utilizes a set of several motions generated by DOC to create a database with the known weights of cost functions. The motion data-set is represented in the low-dimensional space characterizing the weights by using FPCA. The weights of newly observed motion can be estimated by projecting the motion in the FPC space. The synthesizing process is so simple that the method requires very low computing time for estimating the weights from new motion data.

2

Motion Analysis Framework

Figure 1 shows the flowchart for estimating the IOC weight of cost functions from observed motion using FPCA. The whole process is divided into three steps. 1) Generation of the training data-set: the data-set of weights of the cost functions are prepared in a systematic way. The joint angle trajectories are generated by DOC using each prepared weights. Each function will be detailed in Sect. 4. Therefore, the weights of the cost functions need to be scaled according to the magnitude of the cost functions in the data-set. Let the scaled weights and the generated joint angle trajectories be called the training data-set. 2) Low dimensional representation of data: the obtained training data-set is converted into a low-dimensional one using FPCA. The compressed data-set can be expressed as points in the low-dimensional space called FPC space. By analyzing the distribution of the points in the space, it is possible to confirm

Motion Synthesis Using Low-Dimensional Feature Space and Its Application

505

Fig. 1. Flowchart of the proposed weight estimation method

the characteristics or tendencies of the data-set. For example, if the distance between two points is close, the corresponding two data samples have similar characteristics i.e., IOC weights. 3) Weight estimation from low dimensional data-set: for weight estimation, the approximation ratio is needed. This ratio is acquired by calculating the inverse of the distance between the subject data and the obtained training data-set in the low-dimensional space. Using this approximation ratio and weight matrix, it is possible to estimate weights of the subject data. Therefore, blending the training data-set in the low-dimensional space with the approximation ratio, it is possible to generate new motions. Comparing the generated motion data and the subject motion data, it is possible to evaluate accuracy of our weights estimation method.

3

Estimation and Synthesis Method Using FPCA

In Direct Optimal Control (DOC), the optimized motion data Q (Q = [qt q˙t q¨t ]) is obtained by solving the problem of minimizing the set of cost function Ji (Q) with corresponding weight ci under equality constraints h(Q) as follows:

506

S. Shimizu et al.

min

Q =q t ,q˙t ,q¨t

J(Q) =

n 

αi ci Ji (Q)

subject to h(Q) = 0

(1)

i=1

where, qt , q˙t , q¨t are the vectors of angle, angular velocity, and angular acceleration at time instance t, n is the number of cost functions, αi is the scaling number that is determined according to the magnitude of the objective function Ji (Q) after calculating it. Let Qobs be the solution of the optimized trajectory. In IOC, when a motion data Qobs is given, cost function weights c (= [c1 · · · cn ]T ) can be estimated. IOC is expressed as the inverse Karush-KuhnTucker (KKT) problem formula [7]. The KKT Lagrangian L(Qobs ) is defined by the following equation, L(Qobs ) =

nJ 

αi cˆi Ji (Qobs ) +

i=0

nh 

λj hj (Qobs )

(2)

j=0

where, cˆi and λj are the unknown variables in the IOC problem, nJ and nh are the number of corresponding functions J(Qobs ) and h(Qobs ). To archive cˆi , we have to solve the stationary condition about the gradient such that ∇L(Qobs ) = 0. Since the gradient can contain measurement noises and modeling errors, we solve the following optimization problem: min ∇L(Qobs )

subject to cˆ ≥ 0,

cˆ,λ

n 

cˆi = 1

(3)

i=1

Data compression using FPCA converts a multidimensional data into a lowdimensional point. In this paper, Qobs can be represented as: Qobs =

NT 

Q t , Qt =

t=1

NB 

wi bi,t

(4)

i=1

where, NT is the time length of Qobs , Qt ∈ RNJ is the vector of joint angle at time instance t, NJ is the number of joints. bi,t ∈ R is the cubic B-spline function, NB is the number of spline functions in the basis, and wi ∈ RNJ is expressed the cost matrix. Next, the corresponding scalar value f (n) called FPC score is defined as follow: f

(n)

=

NT  t=1

(n)

(n) ξtT Qt ,

ξt =

NB 

θi bi,t

(5)

i=1

where, Qt is the joint angle vector at time sample t of n-th data set, and θi ∈ RNJ refers to the vector of costs of the regression model in Eq. 5. The analysis method of FPCA calculates the basis ξ to maximize the variance of the (n) FPC scores using ξt and Qt .

Motion Synthesis Using Low-Dimensional Feature Space and Its Application

507

After performing FPCA, the relationship between the FPC scores f in the NM dimension FPC space and the coefficients w of the B-spline bases can be expressed simply as follows: f = M (w − w)

(6)

where, M ∈ RNM ×NB NJ is the conversion matrix which can be calculated from ξ and bi,t , and w indicates the averaged coefficients determined by the training data-set. For weight estimation, a weight combination matrix C ∈ RNm Nn is needed. This matrix includes c = [c1 , c2 , · · · cn ] and is denoted as following: ⎡ ⎤ ⎡ ⎤ c11 c12 · · · c1n c1 ⎢ c2 ⎥ ⎢ c21 c22 · · · c2n ⎥ ⎢ ⎥ ⎢ ⎥ (7) C=⎢ . ⎥=⎢ . .. . . .. ⎥ ⎣ .. ⎦ ⎣ .. . . ⎦ . cm cm1 cm2 · · · cmn where, Nm is the number of training data and Nn is the number of cost functions, so C can be regarded as coordinates in Nn dimension space. All cost combinations are converted into FPC score through DOC and FPCA. Let the scaled weights and the generated joint angle trajectories and the FPC score be called the training data-set. It is possible to estimate weights of cost functions from subject motion data ˆ First, the subject motion Q ˆ is converted and plotted in FPC space. Using the Q. training data-set of FPC score, the distance di between the subject data fˆ and neighboring training data fˆi is denoted as fˆ − fˆi . The approximation ratio rˆ are calculated in the following equation:

T  4 1 dˆnk dˆ1 dˆ2 ˆ , ,··· , di = (8) rˆ = di + ε dˆ dˆ dˆ where, nk is the number of neighboring FPC scores and ε is a small value. dˆ is the summation of dˆi . Weight estimating uses the approximation ratio rˆ and the weight vector cˆi . cˆi is in the weight combination matrix C and corresponds to the neighboring FPC score fˆi . cˆ is denoted as: cˆ = cˆi rˆ

(9)

According to Eq. 9, we can estimate the unknown weights by using FPCA instead of IOC.

4

Validation of Variable Estimation of Motion Data

This section evaluates the performance of the proposed method by comparing the obtained results with state of the art IOC. In this paper, the following 3 functions are set as objective functions for human motion analysis [6,18,19].

508

S. Shimizu et al.

Minimum jerk cost motion

Minimum torque cost motion

Fig. 2. 7-joints humanoid robot arm animations for two different sets of weights

(1) Angular jerk: Jdddq = (2) Torque: Jtau =

nd  T  ...2 q d,t

nd  T 

(3) Power: Jpower =

d 2 τd,t

t d nd  T  d

t

(q˙d,t τd,t )2

t

From Eq. 4, the motion data includes only ˙ ... angle values, so angular velocity q, angular acceleration q¨ and angular jerk q are calculated by numerical differentiation. Torque τ is defined by angle, angular velocity, and angular acceleration using the inverse dynamics. The costs vector ci is a three dimensional vector

[c1 ∼ c3 ](0 ≤ ci ≤ 1, ci = 1). This time, the values of (c1 , c2 , c3 ) are changed by 0.01 increment and used for the weight combination matrix. The training data-set has 5151 motion data. A giving reaching motion is adopted as a subject data, and extracted 7-joints humanoid robot arm as a model. The humanoid robot model is HRP-4 [20]. In the training data-set and the subject motion, similar initial and final values such as angle and angular velocity are set. About weight estimation, we created the subject motion data Qsubject with random c1 , c2 , c3 through DOC. The error is defined as the absolute difference between those costs and the estimated costs cˆ. Therefore, using motion synthesis ˆ is created form cˆ and compared with the entire subject motion with FPCA, Q data Qsubject . Figure 2 shows robot motion animations which each kind of the cost function is minimum. These motion data has same initial and final values, but the

Motion Synthesis Using Low-Dimensional Feature Space and Its Application

509

movements are different. In Fig. 4b, the right arm swings back comparing with Fig. 4a. Table 1 shows the errors and the computation times of IOC and proposed method. The errors are the summations of absolute difference between original costs and the estimated costs cˆ. In this Table 1, the error of our proposed method Table 1. Error of weight estimation and computation time for state of the art IOC method and the proposed method

7-joints Humanoid Robot Arm No. IOC Proposed method Time(s) Error Time(s) Error 1 2 3 4 5 6 7 8 9 10

0.7792 0.7013 0.7397 0.7183 0.7894 0.8166 0.7409 0.7215 0.7508 0.7172

Ave 0.7475

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1204 0.2055 0.1209 0.1279 0.1157 0.1198 0.1181 0.1263 0.1200 0.1273

0.0182 0.0028 0.0056 0.0275 0.0053 0.0066 0.0037 0.0152 0.0036 0.0488

0.000 0.1302

0.0137

Fig. 3. 7-joints humanoid robot arm variables of objective functions

510

S. Shimizu et al.

is kept small while the computation time is 3 times faster than state of the art ˆ IOC. Figure 3 shows original motion data Qsubject and the reproduced data Q from the estimated costs cˆ. The solid line is the original motion data Qsubject , ˆ Figure 3 also shows their errors. and the broken line is the reproduced data Q. Objective functions are composed of squares of angular derivatives, so if the accuracy of motion reproduction is low, the errors in Fig. 3 will become large ˆ reproduced the origin Qsubject easily. These errors become are quite small, so Q with high accuracy. From the above results, the proposed method with FPCA is superior to IOC in the point of computation time with practical accuracy.

5

Conclusion

For human-like motion planning, the cost of the motion objective function is the important component to assess the motion characteristic. This cost is acquired by IOC. However, the computation time is long. In this paper, we presented a novel method for estimating the costs of the objective functions of the multiple joint motion with FPCA. Our method calculates approximation ratio between the subject data and the training data-set in the low-dimensional space, and obtains the costs ci with the ratio. The experiments show that the accuracy of our FPCA method is almost same as IOC and its computation time is shorter than IOC. In the experiments, there are only 3 objective functions. In the case of actual analysis subject motion, the number of assumed objective functions is quite large. In our future work, we need to increase the various objective functions such as summation of torque effort and kinetic energy. We also have to consider the number of training data-set. Although the estimation is precise with a large amount of data-set, it is impractical with regard to computation time to prepare. Current data-set has 5151 data, and preparation time takes over 12 h. For whole body motion estimation, the number of training data-set will be regulated carefully.

References 1. Ayusawa, K., Yoshida, E., Imamura, Y., Tanaka, T.: New evaluation framework for human-assistive devices based on humanoid robotics. Adv. Robot. 30(8), 519–534 (2016) 2. Miura, K., Yoshida, E., Kobayashi, Y., Endo, Y., Kanehioro, F., Homma, K., Kajitani, I., Matsumoto, Y., Tanaka, T.: Humanoid robot as an evaluator of assistive devices. In: Proceedings of the 4IEEE International Conference on Robotics and Automation, pp. 671–677 (2013) 3. Yamane, K., Nakamura, Y.: Natural motion animation through constraining and deconstraining at will. IEEE Trans. Vis. Comput. Graph. 9(3), 352–360 (2003) 4. Kuffner, J.-J., Nishiwaki, K., Kagami, S., Inaba, M., Inoue, H.: Motion planning for humanoid robots under obstacle and dynamic balance constraints. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 692–698 (2001)

Motion Synthesis Using Low-Dimensional Feature Space and Its Application

511

5. Yue, H., Mombaur, K.: Bio-inspired optimal control framework to generate walking motions for the humanoid robot iCub using whole body models. Appl. Sci. 8(2), 278 (2018) 6. Flash, T., Hogan, N.: The coordination of arm movements: an experimentally confirmed mathematical model. J. Neurosci. 5(7), 1688–1703 (1985) 7. Lin, J.F.-S., Bonnet, V., Panchea, A.M., Ramdani, N., Venture, G., Kuli´c, D.: Human motion segmentation using cost weights recovered from inverse optimal control. In: IEEE-RAS 16th International Conference on Humanoid Robots, pp. 1107–1113. IEEE (2016) 8. Mombaur, K., Truong, A., Laumond, J.-P.: From human to humanoid locomotion - an inverse optimal control approach. Auton. Robots 28(3), 369–383 (2010) 9. Papadopoulos, A.V., Bascetta, L., Ferretti, G.: Generation of human walking paths. Auton. Robots 40(1), 59–75 (2016) 10. Carreno-Medrano, P., Harada, T., Lin, J.F.-S., Kuli´c, D., Venture, G.: Analysis of affective human motion during functional task performance: an inverse optimal control approach. In: 2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids), pp. 461–468. IEEE (2019) 11. El-Hussieny, H., Abouelsoud, A.A., Assal, S.F.M., Megahed, S.M.: Adaptive learning of human motor behaviors: an evolving inverse optimal control approach. Eng. Appl. Artif. Intell. 50, 115–124 (2016) 12. Besse, P., Ramsay, J.-O.: Principal components analysis of sampled functions. Psychometrika 51(2), 285–311 (1986) 13. Aleotti, J., Caselli, S.: Functional principal component analysis for recognition of arm gestures and humanoid imitation. Int. J. Humanoid Robot. 10(4), 1–25 (2013) 14. Nicol, F.: Functional principal component analysis of aircraft trajectories (2013) 15. Iwasaki, T., Okamoto, S., Akiyama, Y., Yamada, Y.: Principal motion ellipsoids: gait variability index based on principal motion analysis. In: 2020 IEEE/SICE International Symposium on System Integration (SII), pp. 489–494. IEEE (2020) 16. Morishima, S., Ayusawa, K., Yoshida, E., Venture, G.: Whole-body motion retargeting using constrained smoothing and functional principle component analysis. In: IEEE-RAS 16th International Conference on Humanoid Robots, pp. 294–299. IEEE (2016) 17. Shimizu, S., Ayusawa, K., Yoshida, E., Venture, G.: Whole-body motion blending under physical constraints using functional PCA. In: IEEE-RAS 18th International Conference on Humanoid Robots, pp. 280–283. IEEE (2018) 18. Berret, B., Chiovetto, E., Nori, F., Pozzo, T.: Evidence for composite cost functions in arm movement planning: an inverse optimal control approach. PLoS Comput. Biol. 7(10), e1002183 (2011) 19. Safonova, A., Hodgins, J.K., Pollard, N.S.: Synthesizing physically realistic human motion in low-dimensional, behavior-specific spaces. ACM Trans. Graph. 23, 514– 521 (2004) 20. Kaneko, K., Kanehiro, F., Morisawa, M., Akachi, K., Miyamori, G., Hayashi, A., Kanehira, N.: Humanoid robot HRP-4 - humanoid robotics platform with lightweight and slim body. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4400–4407 (2011)

An Analytical Formulation for the Geometrico-Static Problem of Continuum Planar Parallel Robots Federico Zaccaria1(B) , S´ebastien Briot2 , M. Taha Chikhaoui3 , Edoardo Id` a1 , and Marco Carricato1 1

2

DIN, University of Bologna, Bologna, Italy [email protected], {edoardo.ida2,marco.carricato}@unibo.it CNRS, Laboratoire des Sciences du Num´erique de Nantes (LS2N), Nantes, France [email protected] 3 CNRS, Laboratoire TIMC-IMAG, Grenoble, France [email protected]

Abstract. In this paper, we provide an analytical formulation for the geometrico-static problem of continuum planar parallel robots. This formulation provides to an analytical computation of a set of equations governing the equilibrium configurations. We also introduce a stability criterion of the computed configurations. This formulation is based on the use of Kirchhoff’s rod deformation theory and finite-difference approximations. Their combination leads to a quadratic expression of the rod’s deformation energy. Equilibrium configurations of a planar parallel robot composed of two hinged flexible rods are computed according to this new formulation and compared with the ones obtained with state-of-the-art approaches. By assessing equilibrium stability with the proposed technique, new unstable configurations are determined.

1

Introduction

The need of developing robots which can safely interact with their environments has lead robotics researchers to design a new type of robot manipulators named continuum robots [7]. Most of these mechanisms feature a serial architecture, in which rigid bodies and joints are replaced by an assembly of slender rods deformed by wires, electromagnets, fluidic actuators or other types of actuation. Continuum robots, inspired by biological systems such as trunks, tentacles, and snakes, have been used in many applications in which the problem of manipulation in confined, or hard-to-reach workspace, is crucial, like minimally invasive surgery [8,9,11]. The concept of parallel continuum robots was introduced in [5,6]. These first works dealt with the design of continuum Gough-Stewart-like platforms. The robot legs were modeled by using the Cosserat rod theory, and the system of nonlinear ordinary differential equations characterizing the robot’s equilibrium c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 512–520, 2021. https://doi.org/10.1007/978-3-030-58380-4_61

An Analytical Formulation for the Geometrico-Static Problem

513

configurations was solved by a purely numerical approach (shooting method). More recently, several designs of continuum planar parallel robots (CPPRs) have been proposed [1–4], and their geometrico-static problems were analyzed. These mechanisms were modeled by using the Kirchhoff’s rod theory, and the authors proposed a quasi-analytical description of the robot equilibrium configurations. This quasi-analytical form was obtained under the strict condition that external wrenches can only be applied to the platform, and that the deformations are planar. Wrenches applied to other bodies cannot be handled, and spatial robots cannot be modeled. Additionally, according to [1–5], stability analysis of equilibrium configurations is not straightforward. Stability can be assessed according to the numerical method introduced in [12]. In this paper, an alternative formulation stemming from a simpler mathematical framework is introduced. Our approach allows analytical computation of the manipulator’s total potential energy and of its geometrical constraint equations. The minimization of the potential energy subject to geometrical constraints allows for the computation of robot equilibrium configurations. Furthermore, equilibrium stability can be assessed by computing the Hessian matrix of a Lagrangian function [10]. The paper is structured as follows. Section 2 deals with the analytical formulation of the geometrico-static problems and with the computation of the Lagrange function Hessian matrix for generic CPPRs. Section 3 focuses on the geometrico-static analysis of a CPPR composed of two hinged flexible rods. Comparison of results obtained with our approach and with the shooting method are provided. The stability analysis, based on the positiveness of the Hessian matrix, is checked on known stable and unstable equilibria. Further, previously undetected unstable configurations are determined. In the last Section, conclusions are drawn.

2 2.1

Geometrico-Static Modelling of CPPRs Deformation Energy of a Planar Slender Rod

Let us consider an undeformed straight slender rod (Fig. 1(a)), made of an isotropic material and having a constant cross-section along its longitudinal direction. By using the Kirchhoff’s modeling assumption (shear and extensibility of the beam are neglected), the deformation energy, when considering only deformations in the plane (Oxy), is given by: Ve =

1 2



L

E Izz θ2 ds

(1)

0

where E is the material Young’s modulus, Izz is the area moment of inertia of the cross-section, θ = dθ/ds is the derivative of the rotation θ of the rod crosssection (see Fig. 1(a)), L is the rod length, and s is the curvilinear abscissa along the rod.

514

F. Zaccaria et al.

It is possible to approximate the expression of θ by using finite differences [11]. For this, let us discretize the rod into N elements of equal length e = L/N (Fig. 1(b)). The expression of the deformation energy thus becomes: Ve ≈

N 

Vei , where Vei =

i=1

1 2



e

0

E Izz θi2 ds

(2)

with θi being the derivative of the rotation θi of the element i (Fig. 1(b)). The expression of θi can be approximated by using finite differences as follows: θi =

θi − θi−1 , e

(3)

with θ1 = θ0 if the rod is assumed to be clamped at its base (s = 0). s

p(s=0) y

O

undeformed rod p(s) normal to the cross-section at s

x

(s)

elt 1 node i-1 node i

node 1

elt i-1 pi

y

elt i

i

elt N

p(s=L) O

x Fixed frame

(a) Continuous parameterization

node N+1

node i+1

x Fixed frame

(b) Discrete parameterization

Fig. 1. Parameterization of the deformation of a slender rod

Then, introducing Eq. (3) into Eq. (2) leads to: Vei =

1 E Izz (θi − θi−1 )2 2 e

(4)

From Eq. (4), we can notice that the deformation energy is analogous to the energy of a torsional spring of stiffness Keq = E Izz /e located between elements i − 1 and i. Now, considering external constant force fi and moment mi applied at node i1 , the potential energy associated to this loading is equal to: T  (5) Vli = −fiT (pi − p0i ) − mi (θi − θi0 ) = − (pi − p0i )T θi − θi0 wi where (.)0 denotes the undeformed state [11], wi = [fiT mi ]T is the external loading, and pi is the position of node i, which can be computed by the recursive formula:   cos θi pi = pi−1 + e , for i = 2, . . . , N, (6) sin θi with p1 being a known position if the rod is clamped at s = 0. 1

Gravitational loads were neglected for brevity sake, but they can be introduced into Eq. (5) as nodal external actions fi and mi .

An Analytical Formulation for the Geometrico-Static Problem

515

Thus, the total potential energy of the rod, function of rotations θ1 to θN , is: Vrod (θ1 , ..., θN ) =

N 

(Vei + Vli ) + Vl,(N +1)

(7)

i=1

2.2

Potential Energy of CPPRs

Let us consider a CPPR composed of n continuum chains (called legs) (Fig. 2(a)). In this paper, legs will be considered as identical, actuated by a revolute motor at one end (points Ak , k = 1, ..., n) and attached to a rigid platform via passive revolute joints at the other end (points Bk , k = 1, ..., n)2 .

P

B1

P

rigid platform

Bn

y

Bk

leg 1

passive revolute joints A1

motor 1

leg n leg k An

Ak

motor n

motor k

(a) A general CPPR

A1 motor 1

A2

x

motor 2

(b) CPPR made of 2 slender rods

Fig. 2. Schematics of CPPR

The total potential energy of the CPPR is given by: Vtot (qa , θ, p, φ) =

n 

Vrodk (qak , θk , wk ) − f T (p − p0 ) − m (φ − φ0 )

(8)

k=1

where Vrodk is the total potential energy of the leg k, which is a function of the rotations θk = [θk2 . . . θkNk ]T (θkj being the deformation of the element j for leg k, and Nk being the number of elements for the same leg), of the motor variables qa = [qa1 . . . qan ]T (qa1 = θk1 ), and of the nodal loadings wk = T T . . . wkN ] (wki is the nodal wrench on the node i of the leg k), f is a constant [wk1 force exerted on the robot platform, m is a constant moment, p is the location of the force application point P , and φ is the platform’s orientation (Fig. 2(a)). Moreover, θ = [θ1T , . . . , θnT ]T .

2

Other types of joints could be considered as well, but we focus on revolute joints for the sake of brevity.

516

F. Zaccaria et al.

Variables qa , θ, φ and p are related by the following geometric constraints: Φk = pBk (p, φ) − pBk (qak , θk ) = 0

(9)

where pBk is the location of the center of the platform’s revolute joint for the leg k, and (10) pBk (p, φ) = p + R(φ, z)pP Bk , where R(φ, z) is the (planar) rotation matrix of angle φ around the z-axis and pP Bk the constant vector between points P and Bk in the platform frame, and pBk (θk ) can be computed from (6), starting from the known rod extremity located at point Ak . In what follows, we denote as Φ the vector stacking all constraints Φk . 2.3

Geometrico-Static Model for CPPRs

Feasible robot configurations x = [θ T pT φ]T are stable static equilibria for fixed motor positions qa . They appears if and only if the robot internal configuration x leads to a minimum of the potential energy Vtot . Lagrange conditions provide the following characterization of local extrema. Under the condition that ∇x Φ(qa , x) is full rank, x is a local extremum for a fixed motor position qa if and only if there exist multipliers λ ∈ RnΦ (nΦ being the length of the constraint vector Φ) such that (11) ∇x Vtot (qa , x) + ∇x Φ(qa , x) λ = 0, Φ(qa , x) = 0. Note that the expression at the left-hand side of Eq. (11) is the gradient ∇x L(qa , x) of the Lagrangian function L(qa , x) = Vtot (qa , x) + Φ(qa , x)T λ

(12)

The implicit geometrico-static model (11) is a system of m equations (m = mx + nΦ , where mx is the dimension of x) and m + n unknowns. As a consequence, fixing n variables to desired values yields a square system, which generically has a finite number of solutions. The forward geometrico-static problem (FGSP ) consists in fixing the n motor positions qa and computing n controlled coordinates, the uncontrolled coordinates x and the Lagrange multipliers λ so that (qa , x, λ) = (qa , θ, p, φ, λ) is solution to Eq. (11). The inverse geometricostatic problem (IGSP ) consists in assigning the n controlled coordinates (p, φ if n = 3, a subset of it if n < 3, and p, φ and some other variables in θ if n > 3) and computing the corresponding n motor positions qa , the remaining unknown variables in θ, and the Lagrange multipliers λ. 2.4

Equilibrium Stability

At this stage, we do not know if the configuration qa and x solution of the geometrico-static model (11) is stable or not. Without loss of generality, in what follows, we define the stability conditions based on the FGSP equations.

An Analytical Formulation for the Geometrico-Static Problem

517

Consider a set of solutions (q∗a , x∗ , λ∗ ) to Eqs. (11). The stability of the configuration (q∗a , x∗ ) can be checked by verifying the following second-order condition [10]: (13) Hp (q∗a , x∗ , λ∗ ) = ZT HZ  0 in which: – H = ∇x,x L(q∗a , x∗ , λ∗ ) is the Hessian of the Lagrangian L (see (12)) with respect to the variables in x computed for the configuration (q∗a , x∗ , λ∗ ) – Z is the matrix that spans the null space of ∇x Φ(q∗a , x∗ ) computed at (q∗a , x∗ ), i.e. (14) ∇x Φ(q∗a , x∗ )Z = 0 2.5

Solver

It is generally impossible to find a closed-form solution for Eq. (11) because of its non-linearity. Therefore, we use a trust-region algorithm. A multi-start procedure is used in order to identify as many equilibria as possible, i.e. the algorithm is run several times with a random selection of the initial guess.

3

Case Study

In this Section, we study the CPPR composed of two rods presented in [1] (Fig. 2(b)). It is called a RF RF R robot because it is composed of two actuated revolute (R) joints, each being mounted on the ground and attached at one extremity of a flexible rod (F ). Both flexible rods are connected at their tip through a passive revolute (R) joint. Parameters of the rods are: L = 1 m, circular cross-section of radius 1 mm, E = 210 GPa. The distance A1 A2 between the two motors is A1 A2 = 0.5 m.

Computation time [s]

102

101

100

10-1

(a) A stable solution

50 100 200 300 Number of elements

(b) Computation time with respect to the number of elements

Fig. 3. Stable solution and computation time for solving the FGSP, for qa1 = 120◦ , qa2 = 60◦

518

F. Zaccaria et al.

The aim of this study is not to perform a full geometrico-static evaluation of the robot, but to illustrate the main capabilities of our method, namely the simple equilibrium evaluation and the straightforward stability analysis. First, we compute solutions of the FGSP for qa1 = 120◦ , qa2 = 60◦ . We identify a single stable solution (Fig. 3(a)). The evolution of the computational time with respect to the total number of elements for this configuration is provided in Fig. 3(b). Results are provided for a CPU Inter Core i7-6700HQ, 2.60 GHz, 8 Gb of RAM. The convergence time becomes high (>10 s) for a number of elements larger than 100. However, this convergence time is also highly dependent on the initial guess. If the solution is close enough, the algorithm converges in a few iterations. Accuracy of the computed solutions for both the forward and inverse geometrico-static problems is assessed in Fig. 4. We compare our results with those obtained by the geometrically exact shooting method [5] directly applied to the set of differential equations characterizing the robot. Obviously, the accuracy of our method is directly related to the number of elements that we used. In general, a difference of less than 1 mm (for a robot composed of two rods of 1 m) on the prediction of the location of the end-effector is obtained when the number of elements per leg exceeds 100.

Fig. 4. Comparison of our method with respect to the shooting method [5] in terms of robot configuration estimation, as a function of the number of elements Nelts per leg.

Finally, we check the stability of the computed equilibria based on criterion (13). All equilibria identified as stable in [1–4] (Figs. 3 and 4) lead to the positive-definiteness of the Hessian, whereas the unstable equilibrium shown in [2] (Fig. 5(a)), did not satisfy the condition (13). Based on the criterion (13), we are able to find more unstable configurations (Fig. 5(b)) than those identified before. These results show that our analytical formulation is consistent with state-ofthe-art methods in terms of configuration prediction, at the cost of a high number of elements. However, contrary to previous approaches, it has the advantage of providing an analytical formulation of the problem, leading to a straightforward computation of the stability condition. This formulation can be very useful for

An Analytical Formulation for the Geometrico-Static Problem

(a) Detected in [2]

519

(b) Five unstable configurations found when checking criterion (13), for qa1 = 120 deg, qa2 = 60 deg

Fig. 5. Unstable configurations.

the singularity and stability analysis of this type of robots, which is still an open research question.

4

Conclusion

In this paper, we provide an analytical formulation of the geometrico-static problem of CPPRs, leading to an analytical computation of set of equations that must be solved in order to find equilibrium configurations. We also provide a stability criterion. This formulation is based on the use of the Kirchhoff’s rod theory, and the use of finite-difference approximations. This formulation leads to a quadratic expression of the rod deformation energy. The solutions provided by our method were compared with the geometrically exact solutions computed with the shooting method. Results show that, for having a good prediction accuracy, a high number of elements must be used. However, the analytical formulation of the problem, leading to a straightforward computation of the stability condition, is a major advantage, as it can be very useful for the singularity and stability analysis of CPPRs which is still an open research question. In the future, the complete analysis of the geometrico-static performance of several CPPRs, among which the RF RF R robot, will be carried out with our method.

References 1. Altuzarra, O., Caballero, D., Campa, F., Pinto, C.: Forward and inverse kinematics in 2-DOF planar parallel continuum manipulators. In: Proceedings of the European Conference on Mechanism Science (EuCoMeS 2018), Aachen, Germany, pp. 231– 238 (2018)

520

F. Zaccaria et al.

2. Altuzarra, O., Caballero, D., Campa, F.J., Pinto, C.: Position analysis in planar parallel continuum mechanisms. Mech. Mach. Theory 132, 13–29 (2019) 3. Altuzarra, O., Caballero, D., Zhang, Q., Campa, F.: Kinematic characteristics of parallel continuum mechanisms. In: Proceedings of Advances in Robot Kinematics (ARK 2018), Bologna, Italy, pp. 293–301 (2018) 4. Altuzarra, O., Merlet., J.: Certified kinematics solution of 2-DOF planar parallel continuum mechanisms. In: Proceedings of the IFToMM World Congress on Mechanism and Machine Science, Cracow, Poland, pp. 197–208 (2019) 5. Black, C., Till, J., Rucker, D.: Parallel continuum robots: modeling, analysis, and actuation-based force sensing. IEEE Trans. Rob. 34(1), 29–47 (2017) 6. Bryson, C.E., Rucker, D.C.: Toward parallel continuum manipulators. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 778– 785. IEEE (2014) 7. Burgner-Kahrs, J., Rucker, D., Choset, H.: Continuum robots for medical applications: a survey. IEEE Trans. Rob. 31(6), 1261–1280 (2015) 8. Chikhaoui, M.T., Lilge, S., Kleinschmidt, S., Burgner-Kahrs, J.: Comparison of modeling approaches for a tendon actuated continuum robot with three extensible segments. IEEE Robot. Autom. Lett. 4(2), 989–996 (2019) 9. Chirikjian, G.S., Burdick, J.W.: Kinematically optimal hyper-redundant manipulator configurations. IEEE Trans. Robot. Autom. 11(6), 794–806 (1995) 10. Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, Heidelberg (2006) 11. Peyron, Q., Rabenorosoa, K., Andreff, N., Renaud, P.: A numerical framework for the stability and cardinality analysis of concentric tube robots: introduction and application to the follow-the-leader deployment. Mech. Mach. Theory 132, 176–192 (2019) 12. Till, J., Rucker, D.: Elastic stability of Cosserat rods and parallel continuum robots. IEEE Trans. Rob. 33(3), 718–733 (2017)

Static Analysis and Design of Extendable Mechanism Inspired by Origami Structure Based on Non-overconstrained Kinematically Equivalent Mechanism Reiji Ando(B) , Hiroshi Matsuo, Daisuke Matsuura, Yusuke Sugahara, and Yukio Takeda Tokyo Institute of Technology, Meguro-Ku, Tokyo 152-8552, Japan {ando.r.af,matsuo.h.ae,matsuura.d.aa,sugahara.y.aa, takeda.y.aa}@m.titech.ac.jp

Abstract. When a deployable origami structure is kinematically modelled as a link mechanism by replacing the creases by revolute joints, the obtained mechanism is generally overconstrained. In practice, devices and machines originated from overconstrained mechanisms need in-situ adjustments for assembly, and even encounter difficulty in assembly since they are sensitive to geometric errors. In some cases, smooth motion cannot be generated. To develop a high performance extendable arm inspired by an overconstrained origami while solving the problems above, this paper introduces the non-overconstrained kinematically equivalent mechanism (NO-KEM) of an origami structure in the design. Generation and confirmation scheme of NO-KEM is introduced based on the velocity relationship. Numerical examples on the static analysis of NO-KEMs of an origami as a determinate problem are shown and a practical design of an extendable arm based on the concept of NO-KEM having superior static characteristics is shown. Finally, experimental observations based on this design are shown to support the proposed method. Keywords: Origami · Origami Spring · Extendable mechanism · Design · Statics · Overconstrained mechanism · Kinematically equivalent mechanism

1 Introduction There are strong demands for extendable arms used for assisting human in factories, construction sites, medical/care facilities and so on, and a lot of devices inspired by origami structures for such purposes have been developed [1–3]. Main features of origami structures attracting great attentions are expected to be the compactness, lightness, and diversity of configurations. When the devices are fabricated by thin materials such as paper, plastic film, and elastomer, these advantageous features can be relatively easily realized though several problems still exist due to the effect of thickness of the materials. However, when devices are built by rigid components to get higher load capacity inspired by the origami structures, several practical problems become prominent. © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 521–529, 2021. https://doi.org/10.1007/978-3-030-58380-4_62

522

R. Ando et al.

When a deployable origami structure is kinematically modelled as a link mechanism by replacing the creases by revolute joints, the obtained mechanism is generally overconstrained. Though the input-output relationships in motion and force of such overconstrained mechanisms can be theoretically derived, the force analysis of the overall mechanism is indeterminate. In practice, devices and machines originated from overconstrained mechanisms need in-situ adjustments for assembly, and even encounter difficulty in assembly. In some cases, smooth motion cannot be generated. As for the origamiinspired mechanism, unfolding motion, one of its typical advantageous features, cannot be always achieved due to the geometric errors in the device. To solve these problems caused by overconstraints, force analysis of an overconstrained mechanism inspired by origami structures can be theoretically performed by considering kinematic constraints and elastic deformation [4, 5]. However, since their solutions are based on optimization and sensitive to geometric errors, such results are not useful in the practical design. Starting from the overconstrained mechanism, by adding passive joints to reduce the overconstraints, non-overconstrained mechanisms which perform kinematically equivalent motion to that of the original overconstrained mechanism can be obtained. For an origami inspired overconstrained mechanism composed of revolute joints, such nonoverconstrained kinematically equivalent mechanisms (NO-KEMs) can be generated by replacing appropriate number of revolute joints by cylindrical joints. The kinematically identical manipulator (KIM), which performs the same kinematics to but has different architecture from the overconstrained parallel manipulator with 3-UPR architecture, has been proposed in [6]. Six KIMs of the manipulator has been clarified and their compliance characteristics have been compared. The static characteristics of these NO-KEMs (or NO-KIMs) differ from each other though input-output relationship in force is the same among them. Though some other works than [6] have been done, such as on the static analyses based on the concept of NO-KEM [7, 8], design and fabrication of such mechanisms based on NO-KEM have never been investigated. In this paper, to support the development of a high performance extendable arm inspired by an overconstrained origami structure while solving the problems coming from overconstraints, NO-KEM of an origami structure is introduced. Generation and confirmation scheme of NO-KEMs is presented. Numerical examples on the static analysis of NO-KEMs of an origami structure are shown and a practical design of an extendable arm having superior static characteristics is shown. Finally, experimental observations based on this design is presented to show the effectiveness of the proposed method.

2 Non-overconstrained Kinematically Equivalent Mechanism 2.1 Target Mechanism An extendable arm inspired by Origami Spring [9] shown in Fig. 1 has been developed as shown in Fig. 2 [10]. This paper focuses on one segment of this arm shown in Fig. 3 as a demonstrative example. Its kinematic model obtained by replacing the creases by revolute joints is shown in Fig. 4. Since it is basically composed of 6R (R: revolute joint) single-loop spherical chains with three degrees of freedom (DOF) [11], it is an overconstrained mechanism. In some cases, for example, we experienced difficulty in achieving unfolding due to the geometric errors.

Static Analysis and Design of Extendable Mechanism

Fig. 1. Motion of Origami Spring [10]

(a) Contracted

(b) Extended

523

Fig. 2. Motion of Extendable arm [10]

(c) Unfolded

Fig. 3. One segment of extendable arm shown in Fig. 2. [10] End effector

Fig. 4. Original mechanism of one segment

2.2 Non-overconstrained Kinematically Equivalent Mechanism (NO-KEM) for Deterministic Static Analysis The target mechanism shown in Fig. 4 consists of 36 links and 43 revolute joints. There are nine serial kinematic chains arranged in parallel between the stationary link and Pe , the reference point of the end effector. ri (i = 1 . . . 43) represent revolute joints, three of which are assigned as active joints since this mechanism has three DOF [10]. Applying the free-body-diagram approach, 210 equations to represent the equilibrium on forces and moments of all links except the stationary link are derived. As for the joint forces and moments, 218 (= 40 × 5 + 3 × 6) components exist since there are 40 passive revolute joints and 3 active revolute joints. Due to the difference between the numbers of equations and unknown variables as mentioned above, the inverse statics becomes indeterminate. This is due to the nature of overconstraints. If sufficient number of equations to represent

524

R. Ando et al.

the closed-loop condition of the mechanism with consideration to the elastic deformation of the links are derived and solved together with the equilibrium equations mentioned above, all the joint forces and moments can be obtained as the solution of static analysis [12]. This approach is based on the assumption that the mechanism is an overconstrained mechanism. Therefore, the analysis result is not accurate since the actual joint forces are sensitive to the geometric errors. On the other hand, for an overconstrained mechanism, there exist several nonoveconstrained mechanisms which perform kinematically equivalent motion to the original overconstrained mechanism. Such non-overconstrained mechanisms can be obtained by reducing the constraints imposed by joints until the overconstraint is resolved while all the relative motions between links are kept the same. If this process is appropriate, the resultant mechanism is called “non-overconstrained kinematically equivalent mechanism (NO-KEM)” in this paper. For the mechanism shown in Fig. 4, NO-KEMs can be obtained by replacing appropriate set of eight revolute joints having 5 joint force components each by cylindrical joints having 4 joint force components each. As the result, the total number of joint forces/moments as the unknown variables becomes 210 for an NO-KEM, the same as the number of equations in the inverse statics so that the static analysis is deterministic. A method to confirm whether the resultant mechanism is NO-KEM or not will be presented in the next section. 2.3 Generation and Confirmation of NO-KEM The velocity of the end effector vPe = [ω; v] (ω: angular velocity, v: linear velocity) is written in terms of each serial chain between the stationary link and the end effector such as:       sr22 sr31 sr13 (1) ωr13 + −−−→ ωr22 + −−−→ ωr31 = vPe , −−−→ Pr13 Pe × sr13 Pr22 Pe × sr22 Pr31 Pe × sr31 which corresponds to the kinematic chain indicated by a brown solid line in Fig. 4. Here, Pi , si , and ωi represent the position, axis and velocity of joint i(i = r13 , r22 and r31 ). Since there are nine kinematic chains in parallel between the stationary link and the end effector as shown by brown lines in Fig. 4, there exist nine equations to represent the by velocity of the end effector [10]. When a revolute joint r13 is replaced  a cylindrical  0 vr13 to Eq. (1), joint as an example, the velocity relationship is obtained by adding sr13 where vr13 is the linear velocity of the cylindrical joint r13 . If eight out of 40 passive revolute joints are replaced by cylindrical joints, the following equation is obtained by eliminating vPe as a set of loop-closure equations of this mechanism: Aa ωa + Ap ωp = 0,

(2)

where ωa , ωp , Aa and Ap represent 3 dimensional vector representing active joints’ velocity, 48 dimensional vector representing passive joints’ velocity (32 passive revolute joints and 8 passive cylindrical joints), 48 × 3 coefficient matrix and 48 × 48 coefficient matrix, respectively. In order for the mechanism obtained by replacing eight passive revolute joints by passive cylindrical joints to be an NO-KEM of the original mechanism, the following two conditions should be satisfied:

Static Analysis and Design of Extendable Mechanism

525

(a)   dim KerAp = 0

(3)

  vrj = 0 rj : eight passive cylindrical joints

(4)

(b)

Based on the discussions above, the following procedure is proposed to generate and confirm all the possible NO-KEMs of the original overconstrained mechanism. (1) Select three active joints from the 43 revolute joints so that these active joints are independent. For the mechanism shown in Fig. 4, 12,277 sets of three active joints exist as effective ones. (2) For each set of active joints obtained in (1), generate all the possible mechanisms by replacing eight passive revolute joints by passive cylindrical joints. (3) Select all the mechanisms which satisfy the conditions shown as Eqs. (3) and (4) as NO-KEMs. For the mechanism with joints r4 , r5 and r39 being the active joints, 4,469,136 NO-KEMs have been obtained in total.

3 Design of NO-KEM Having Superior Static Characteristics As mentioned in Introduction, the static characteristics of NO-KEMs are different from each other, depending on their structures: selection of the passive revolute joints replaced by cylindrical joints. In this section, a design of NO-KEM is conducted taking an extendable overconstrained mechanism inspired by origami structure shown in Fig. 4 having active joints at r4 , r5 and r39 as an example. The shape of nine lower blue links and the upper red links are changed when one segment is contracted or extended, so the center of the shape of upper links depends on a configuration of the mechanism. When one segment is fully extended, the nine upper links forms a regular nonagon as shown in Figs. 5(a) and 5(b), and the center of the inscribed circle of the nonagon is located very close to the center of the upper links for all configurations. The end effector was designed so that the length of the yellow bar in Figs. 5(b) and 5(c) is the same as the radius of the inscribed circle of the nonagon. External load Fex is supposed to be applied at the tip of the yellow bar (green point in the figure) along the vertical direction.

(a) Fully extended

(b) Length of yellow bar Fig. 5. Definition of end effector

(c) Shape of end effector

526

R. Ando et al. An end effector under 2nd force condition

An end effector under 3rd force condition An end effector under 1st force condition

(a) 1st load condition

(b) Location of end effector

Fig. 6. Three cases of location of output link (external load condition)

(a) Left section

(b) Center section

(c) Right section

Fig. 7. Three sections of the mechanism

Three cases of the locations of the end effector are considered in this section, as shown in Fig. 6. By changing the location of the end effector, different external load conditions can be considered. For the 1st case, the center link of the red upper links is assigned as the end effector, the leftmost and rightmost links for the 2nd and 3rd cases. To clarify the joints which mainly support the external load and to make the determination of the appropriate architecture of NO-KEM less complicated with consideration to their static characteristics, the load support statuses of the three sections of the target mechanism shown in Fig. 7 were separately evaluated. Based on the free-body-diagram approach, static analyses of 50 selected NO-KEMs including mechanisms considered later for the three cases were conducted and the results are summarized in Table 1. This table shows that joints mainly supporting external load depend on the location of the application of external load (end effector). Then, we evaluated the static performance of NO-KEMs for each load condition, and the appropriate architecture of NO-KEM was clarified. In the performance evaluation, the following evaluation function Ef was used, the value of which becomes smaller when its static characteristics is superior, where the magnitude of each joint force was calculated as the average for all the poses in the reachable workspace. Table 1. Joints that support forces or moments under each load condition. ◦ means that joints in the corresponding section mainly support. Joints in left section 1st load condition 2nd load condition 3rd load condition

Joints in center section

Joints in right section

Static Analysis and Design of Extendable Mechanism

(a) Contracted

(b) Extended

527

(c) Unfolded

Fig. 8. Designed prototype of NO-KEM

⎛ ⎞

f r 2 Ef = ⎝ k 2 ⎠ (k = 1 · · · 43) |Fex |

(5)

Under the 1st load condition, Ef is dominated by the arrangement of cylindrical joints in the center section. And, as shown in Table 1, the center section always supports external load. Then, smaller number and symmetrical arrangement of cylindrical joints in the center section was expected appropriate. We decided to symmetrically arrange two cylindrical joints in the center section because we could not find any NO-KEMs having zero cylindrical joints in the center section. Values of Ef for the five possible NO-KEMs, in which the other six cylindrical joints are distributed in both right and left sections to satisfy the conditions shown as Eqs. (3) and (4), are shown in Table 2. From this table, we decided to arrange cylindrical joints at r21 and r23 . Table 2. Ef under the 1st load condition

Under the 2nd load condition, Ef is dominated by the arrangement of cylindrical joints in center and left sections. Since the arrangement of cylindrical joints in the center section has been already determined and symmetrical arrangement of cylindrical joints in the mechanism is preferable, we investigated the arrangement of three cylindrical joints in the left section. As the result, we decided to arrange cylindrical joints at r18 , r19 and r20 in the left section. In the similar way, we decided to arrange cylindrical joints at r24 , r25 and r26 in the right section under the 3rd load condition. As shown in this example, it is found that static characteristics of NO-KEMs greatly depend on the arrangement of cylindrical joints. Based on the results above, we designed an NO-KEM, in which cylindrical joints are arranged at rl (l = 18 · · · 21, 23 · · · 26). Cylindrical joints were designed by removing a constraint along the axis of revolute joints. Through prototyping and experimental observations as shown in Fig. 8, we confirmed that this prototype performs kinematically equivalent extending and contracting motions to the original mechanism and unfolding was achieved easier than the original overconstrained mechanism.

528

R. Ando et al.

4 Conclusion In this paper, to develop a high performance extendable arm inspired by an overconstrained origami structure by solving the problems coming from overconstraints, nonoverconstrained kinematically equivalent mechanisms (NO-KEMs) of an origami structure were introduced. A demonstrative example through design and prototyping was shown. The conclusions are summarized as follows. (1) By replacing an appropriate number of revolute joints by cylindrical joints and setting conditions based on velocity analysis, NO-KEMs of an overconstrained origami structure were successfully obtained. (2) Static characteristics of NO-KEMs are different according to the architecture of NO-KEM. An NO-KEM with superior static characteristics was designed, and a prototype based on it showed that the problems coming from overconstraints were solved by the proposed method. Future work includes experimental performance evaluation, analysis of multiple segments by applying the proposed method in this paper, and design and development of an extendable arm composed of multiple segments with superior static characteristics. Acknowledgments. This work was supported in part by the Grants-in-Aid for JSPS Fellows 18J21466 and in part by NSK Ltd.

References 1. Jeong, D., Lee, K.: Design and analysis of an origami-based three-finger manipulator. Robotica 36(2), 261–274 (2018) 2. Banerjee, H., Pusalkar, N., Ren, H.: Single-motor controlled tendon-driven peristaltic soft Origami robot. J. Mech. Robot. 10(6), 064501 (2018) 3. Fathi, J., Oude Vrielink, T.J.C., Runciman, M.S., Mylonas, G.P.: A deployable soft robotic arm with stiffness modulation for assistive living applications. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2019, pp. 1479–1485. IEEE Press, Montreal (2019) 4. Qiu, C., Zhang, K., Dai, J.S.: Repelling-screw based force analysis of Origami mechanisms. J. Mech. Robot. 8(3), 031001 (2016) 5. Jianguo, C., Xiaowei, D., Yuting, Z., Jiang, F., Ya, Z.: Folding behavior of a foldable prismatic mast with Kresling Origami pattern. J. Mech. Robot. 8(3), 031004 (2016) 6. Hu, B.: Kinematically identical manipulators for the Exechon parallel manipulator and their comparison study. Mech. Mach. Theory 103, 117–137 (2016) 7. Li, W., Angeles, J.: Full-mobility three-CCC parallel-kinematics machines: kinematics and isotropic design. J. Mech. Robot. 10(1), 011011 (2018) 8. Bagci, C.: Static force analysis using 3 × 3 screw matrix, and transmission criteria for space mechanisms. J. Eng. Ind. 93(1), 90–101 (1971) 9. Benyon, J.: Spring into Action. BOS Magazine 142, British Origami Society (1990) 10. Matsuo, H., Asada, H.H., Takeda, Y.: Design of a novel multiple-DOF extendable arm with rigid components inspired by a deployable Origami structure. Robot. Autom. Lett. 5(2), 2730–2737 (2020)

Static Analysis and Design of Extendable Mechanism

529

11. Matsuo, H., Matsuura, D., Sugahara, Y., Takeda, Y.: Modelling and displacement analysis of origami spring considering collision and deformation of components. In: Proceedings of the 15th IFToMM World Congress on Mechanism and Machine Science, pp. 329–336. Springer, Krakow (2019) 12. Takeda, Y., Funabashi, H., Sasaki, Y.: Development of a spherical in-parallel actuated mechanism with three degrees of freedom with large working space and high motion transmissibility (Evaluation of motion transmissibility and analysis of working space). JSME Int. J. Ser. C 39(3), 541–548 (1996)

Function Approximation Technique Based Immersion and Invariance Control for an Underactuated Tower Crane System Yang Bai(B) and Mikhail Svinin Department of Information Science and Engineering, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan {yangbai,svinin}@fc.ritsumei.ac.jp

Abstract. In this paper, a function approximation technique based immersion and invariance (FATII) method is proposed for the control of an underactuated tower crane system. Firstly, the tower crane system is restructured as the combination of an auxiliary system and a variation term from the original system. The variation term is treated as a timevarying uncertainty and parameterized by a group of weighted chosen basis functions. These weights are estimated at every time instant and the change of the estimates is governed by an update law. The update law is defined based on an immersion and invariance approach such that both the system state and the estimation error converge to zero. The asymptotic stability of the proposed method is established and its feasibility is verified under simulations.

Keywords: Robust adaptive control crane system

1

· Underactuated system · Tower

Introduction

The studies of tower crane systems have drawn increasing attentions in recent decades. To control these systems, various methods have been proposed in the literature [12]. Open loop schemes [1,11,17] are easy to implement and extra sensors are not necessary, but they work only under an ideal condition in absence of uncertainties. Feedback control schemes have been developed [4,9,13,16] which are robust to external uncertainties while they commonly require the knowledge of the system model. To deal with problem, intelligent controls [6,10,14] have been introduced such as the fuzzy control and the neural network based control. Controllers of this type are designed directly from the input-output data, bypassing the modelling step. However, the control process relies on the human experience and the asymptotic stability of these methods is difficult to prove. This research was supported, in part, by the Japan Science and Technology Agency, the JST Strategic International Collaborative Research Program, Project No. 18065977. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 530–537, 2021. https://doi.org/10.1007/978-3-030-58380-4_63

FATII Control for an Underactuated Tower Crane System

531

To feature both the stability and the adaptiveness, in this paper, we propose the FATII control method which is robust to rather large uncertainties. On the other hand, compared with the fuzzy or learning based approaches, the asymptotic stability of the FATII method can be proved. The FATII method is based on the function approximation technique (FAT) [2,3,7]. It reconstructs the time-varying uncertainties in the control systems as a group of basis functions weighted by unknown constant parameters. Then, an immersion and invariance (I&I) approach is utilized to eliminate the effect of the unknown parameters to the control system. The rest of this paper is organized as follows. First, in Sect. 2 we introduce the mathematical description of an underactuated tower crane system. Next, in Sect. 3 we illustrate the process for constructing the FATII controller, and establish its asymptotic stability. The feasibility of the proposed control method is verified under simulations in Sect. 4. Finally, conclusions are drawn in Sect. 5.

2

Mathematica Model

x, F

1 2

Fig. 1. Model of an underactuated tower crane system.

An underactuated 4-DOF tower crane (see Fig. 1) is modeled by [15] ˙ = F (q)u, M (q)q¨ + N (q, q) where the components of the symmetric matrix M are   m11 = J + (mp + mt )x2 + mp (S12 C22 + S22 )l2 + 2xlC2 S1 , m22 = mp + mt , m33 = mp l2 C22 , m44 = mp l2 , m12 = −mp lS2 , m13 = −mp l2 C1 C2 S2 , m14 = mp l(C2 x + lS1 ), m23 = mp lC1 C2 , m24 = mp lS1 S2 , m34 = 0,

(1)

532

Y. Bai and M. Svinin

the components of N are

 ˙ 1 − S2 (2φS ˙ 1 + θ˙2 )xθ˙2 + 2S1 C2 x˙ φ˙ n1 = 2(mp + mt )xx˙ φ˙ + mp l 2C1 C2 xφθ  + l sin(2θ1 )C22 φ˙ θ˙1 + lS1 S2 C2 θ˙12 + lC22 sin(2θ2 )φ˙ θ˙2 + 2lC1 S22 θ˙1 θ˙2 ,   n2 = (mp + mt )xφ˙ 2 + 2mp lC1 S2 θ˙1 θ˙2 + mp lC2 S1 (φ˙ 2 + θ˙12 + θ˙22 ) + 2φ˙ θ˙2 ,   n3 = −mp lC1 C2 (x + lS1 C2 )φ˙ 2 + 2l2 C2 (C1 C2 φ˙ + S2 θ˙1 )θ˙2 − glS1 C2 ,   n4 = mp l 2C2 x˙ φ˙ + (xS1 S2 − lC12 S2 C2 )φ˙ 2 + 2lC1 C22 φ˙ θ˙1 + lθ˙12 S2 C2 + gC1 S2 ,

and F4×2 = [I|0] . Here q = (φ, x, θ1 , θ2 ) and u = (M, F ) respectively denote the output and the input vectors where φ denotes the jib slew angle, x denotes the trolley translation displacement, θ1 and θ2 are used to depict the payloadfs swing, M denotes the slew control toque, and F is the translation control force, S1 , S2 , C1 , C2 represent sin θ1 , sin θ2 , cos θ1 , cos θ2 , and J denotes the moment of inertia of the jib, mt represents the trolley mass, mp is the payload mass, L is the suspension rope length, g is the gravitational constant.

3

Controller Design Process

3.1

Control Problem Statement

Based on (1), we construct a feedback controller for the underactuated tower crane system. By introducing, in the spirit of the sliding mode control, the following variable σ = q˙ + Λq, (2) where Λ is a constant positive-definite Hermite matrix. Note that through this design, when σ approaches zero, q converges to zero as well. From (2) one obtains ˙ σ˙ = q¨ + Λq.

(3)

˙ + M (q)−1 F (q)u, and substituting it Rearranging (1) as q¨ = −M (q)−1 N (q, q) into (3) gives the state equation σ˙ = f (σ) + G(σ)u + ξ,

(4)

˙ G = M −1 F , and ξ represents the system uncerwhere f = −M −1 N + Λq, tainties. To square the control system (4), one introduces an auxiliary input u∗ ∈ Rm such that (5) u = G∗ u∗ , where the auxiliary matrix G∗ is chosen to be a full rank n × m matrix. Then the control system (4) is rewritten as

where

σ˙ = u∗ + d,

(6)

d(σ, t, u∗ ) = f + (GG∗ − I)u∗ + ξ.

(7)

FATII Control for an Underactuated Tower Crane System

533

For the restructured system, the number of the inputs equals that of the states. The methods for the design of G∗ are specified in Appendix. Note that an essential condition for the selection of matrix G∗ is that the restructured system (6) must be controllable. Through the above rearrangement, the original system (4) is restructured as the combination of two parts, a trivial linear system σ˙ = u∗ referring to the auxiliary system, and d, which can be viewed as an uncertainty term to the auxiliary system. Thus the original problem is reformulated to the adaptive control problem for a linear system with time-varying uncertainties, which is stated as constructing u∗ such that lim σ = 0, with d unknown. t→∞

3.2

FATII Based Controller Design

To tackle the stated problem, the controller design is under an FAT based framework. We utilize the weighted basis functions to approximate d in the control system (6) at each time instant as d(σ, t, u∗ ) =

N 

di ψi (σ, t) + ,

(8)

i=0

where di is constant and ψi consists of σ and t, and , referring to the approximation error, describes the deviation between the uncertainty d and the weighted basis functions. Note that when N → ∞, the approximation error  would vanish. By selecting enough number of the basis functions,  can be assumed bounded [5,7,8]. Several candidates for the basis function ψi can be selected to approximate the nonlinear functions. In this paper, we select the Fourier series [8] as the basis functions. Note that the control problem requires the unknown plant parameters di in (8) to be identified. For this purpose, these plant parameters di at each time instant t are estimated by dˆi (t), referring to the control parameter. In conventional adaptive control algorithms, the control parameter dˆi (t) is not guaranteed to converge to the actual value. The error between the actual parameters and parameter estimation can produce large uncertainty, which would deteriorate the control accuracy and lead to an undesired transient response of the closed loop system. To converge both the system state and the estimation error, we propose the FATII method based on the I&I approach. Define in the extended space (σ, dˆj ) the manifold (9) Mi = {(σ, dˆi ) ∈ Rn | di − dˆi − βi = 0}, where βi is a continuous function to be specified. The motivation for this definition is described as follows. By substituting (8) into (6), the dynamics of the system restricted to the manifold Mi (provided it is invariant) is described by σ˙ = u∗ +

N  i=0

(dˆi + βi )ψi .

(10)

534

Y. Bai and M. Svinin

˙ Note from (10), the unknown vector di is excluded from the expression of σ, which is important in the controller design process since di cannot appear in the control law. However, (10) is equivalent to (6) only when the system dynamics stay in the manifold Mi . By defining the off-the-manifold variable (11) zi = di − dˆi − βi , where zi ∈ Rn×1 and βi ∈ Rn×1 , zi = 0 implies that the system dynamics stay in the manifold Mi . With the off-the-manifold variable, the state equation is transformed to N    zi + dˆi + βi ψi , σ˙ = u∗ + (12) i=0

N where i=0 zi ψi represents the estimation error of the system uncertainty. Thus, the original control problem can be restated as constructing u∗ such that both the system state and the estimation error converge to zero. For the constructionof u∗ , the Lyapunov candidate function can be formuN lated as V = 12 (σ  σ + i=0 zi zi ), the derivative of which is calculated as     ˆ +B ψ V˙ = σ  u∗ + Z + D +

N 

zi

i=0



    ˆ˙ i − ∂βi − ∂βi u∗ + Z + D ˆ +B ψ , −d ∂t ∂σ

(13)

ˆ = [dˆ0 | dˆ1 | ... | dˆN ], B = [β0 | β1 | ... | βN ], where Z = [z0 | z1 | ... | zN ], D  and ψ = [ψ0 ψ1 ... ψN ] . The design of the auxiliary controller u∗ requires to render the derivative of the Lyapunov candidate function (13) negative semi-definite. For this purpose, one propose the following FATII controller u∗ = −Kσ −

N 

ˆ˙ i = −σ ψ˙ i + Kσψi . (dˆi + σψi )ψi , d

(14)

i=0

Theorem 1. The closed loop system, formulated by the plant (10) and the controller (14), is asymptotically stable. Proof. After substituting (10) and (14) into the derivative of the Lyapunov candidate function, one obtains

 N  N  N     ˙ V = −σ zi ψi − zi ψi zi ψi Kσ − i=0

i=0

i=0

 N  N    1  1   = −σ Kσ + σ σ − σ σ + σ zi ψi − zi ψi zi ψi 4 4 i=0 i=0 i=0 2 N  1 1 zi ψi − σ + σ22 . ≤ −λmin (K)σ22 − 2 4 

i=0

N 

2

FATII Control for an Underactuated Tower Crane System

535

The selection of λmin (K) > 14 renders the derivative of the Lyapunov candidate function V˙ negative semi-definite. By using  the Barbalat’s lemma, one obtains not only limt→∞ σ = 0, but N also limt→∞ i=0 zi ψi = 0, implying that both the state and the uncertainty estimation converge to the desired and the actual uncertainty respectively. The actual control u can be calculated from (5).

4

Case Study

In the simulation, both parametric and nonparametric uncertainties are taken into account. For parametric uncertainties, the parameters for the system dynamics are chosen as mt = 3.5 kg, J = 6.8 kg·m2 , mp = 0.5 kg, and L = 0.5 m. In the controller design, the parameters are set as mt = 4 kg, J = 7 kg · m2 , mp = 0.8 kg, and L = 0.7 m. The nonparametric uncertainty in (1) is selected as ˙ φ, ˙ 0, Ff 1 tanh(ξx x) ˙ + Mf 2 |φ| ˙ + Ff 2 |x| ˙ x), ˙ ξ = (0, Mf 1 tanh(ξφ φ)

(15)

Crane displacement

x

Pendulum angle

where Mf 1 , Ff 1 , Mf 2 , Ff 2 , ξφ , and ξx are friction parameters, ξφ and ξx are determined by the payload masses.

Time (s)

1 2

Time (s)

Fig. 2. Trajectories for the crane displacements (left) and the pendulum angles (right) of a tower crane system under the FATII control.

Time (s)

u1 u2

Input signal

Estimation error

e1 e2 e3 e4

Time (s)

Fig. 3. Uncertainty estimation (left) and the input signals (right) under the FATII control.

536

Y. Bai and M. Svinin

In the simulation, the friction parameters are designed as Mf 1 = Ff 1 = 4.4, ξφ = ξx = 0.01, and Mf 2 = Ff 2 = 0.5. The auxiliary input is design as (14). The gain matrix K in the controller is selected as diag(1, 2, 9, 12) and Λ = diag(17, 17, 10, 12). It can be seen that the state σ converges to zero and therefore, q converges to zero. Here, q represents for the difference between the actual and desired values of x(t), φ(t), θ1 (t), and θ2 (t). The convergence of q implies that the system variables approach the desired values, as illustrated in Fig. 2. The initial conditions of x(t), φ(t), θ1 (t), and θ2 (t) are specified as 0 m, 0 rad, 0.5 rad, 0.5 rad, and the desired values are respectively 0.5 m, 0.5 rad, N 0 rad, 0 rad. The estimation error (e1 (t), e2 (t), e3 (t), e4 (t)) = d − i=0 di ψi between the auxiliary and the actual systems converge to zero, as illustrated in Fig. 3, where i = 1, 2, 3, 4. The initial values of the control parameters dˆi (t) are chosen to be zero. The input signals are also shown by Fig. 3.

5

Conclusions

The FATII control method has been proposed for a class of tower crane systems. The asymptotic stability has been established for the closed loop system formulated by the tower crane system and the constructed controller. The feasibility of the proposed control method has been verified under simulations. The FATII method is based on a robust adaptive approach (FAT) and therefore can reject the effect of the system uncertainties or external disturbances to the control system. Unlike model-free methods such as the soft computing techniques, the stability for systems under the FATII control has been well established. The following issues need to be clarified in the future work. Firstly, a unified way of selecting G∗ needs to be developed. Secondly, we chose the Fourier functions to for estimating the variation term d. However, the advantages and disadvantages for this type of basis function are not analyzed and more candidates need to be investigated. Thirdly, more systems will be tested under the FATII based control and experimental works will also be conducted.

References 1. Abdullahi, A.M., Mohamed, Z., Abidin, M.Z., Buyamin, S., Bature, A.A.: Outputbased command shaping technique for an effective payload sway control of a 3D crane with hoisting. Trans. Inst. Meas. Control 39(10), 1443–1453 (2017) 2. Bai, Y., Svinin, M., Yamamoto, M.: Adaptive trajectory tracking control for the ball-pendulum system with time-varying uncertainties. In: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 2083–2090, September 2017 3. Bai, Y., Svinin, M., Yamamoto, M.: Function approximation based control for non-square systems. SICE J. Control Meas. Syst. Integr. 11(6), 477–485 (2018) 4. Bak, M., Hansen, M., Karimi, H.: Robust tool point control for offshore knuckle boom crane. IFAC Proc. 44(1), 4594–4599 (2011). 18th IFAC World Congress

FATII Control for an Underactuated Tower Crane System

537

5. Chen, H.Y., Huang, S.J.: A new model-free adaptive sliding controller for active suspension system. Int. J. Syst. Sci. 39(1), 57–69 (2008) 6. Duong, S.C., Uezato, E., Kinjo, H., Yamamoto, T.: A hybrid evolutionary algorithm for recurrent neural network control of a three-dimensional tower crane. Autom. Constr. 23, 55–63 (2012) 7. Huang, A.C., Chen, Y.C.: Adaptive sliding control for single-link flexible-joint robot with mismatched uncertainties. IEEE Trans. Control Syst. Technol. 12(5), 770–775 (2004) 8. Huang, A.C., Kuo, Y.S.: Sliding control of non-linear systems containing timevarying uncertainties with unknown bounds. Int. J. Control 74(3), 252–264 (2001) 9. Jaafar, H.I., Hussien, S.Y.S., Ghazali, R., Mohamed, Z.: Optimal tuning of PID+PD controller by PFS for gantry crane system. In: 2015 10th Asian Control Conference (ASCC), pp. 1–6, May 2015 10. Lee, L.H., Huang, P.H., Shih, Y.C., Chiang, T.C., Chang, C.Y.: Parallel neural network combined with sliding mode control in overhead crane control system. J. Vib. Control 20(5), 749–760 (2014) 11. Maghsoudi, M., Mohamed, Z., Tokhi, M., Husain, A., Abidin, M.: Control of a gantry crane using input-shaping schemes with distributed delay. Trans. Inst. Meas. Control 39(3), 361–370 (2017) 12. Ramli, L., Mohamed, Z., Abdullahi, A.M., Jaafar, H., Lazim, I.M.: Control strategies for crane systems: a comprehensive review. Mech. Syst. Signal Process. 95, 1–23 (2017) 13. Sano, S., Ouyang, H., Yamashita, H., Uchiyama, N.: LMI approach to robust control of rotary cranes under load sway frequency variance. J. Syst. Des. Dyn. 5(7), 1402–1417 (2011) 14. Smoczek, J.: Fuzzy crane control with sensorless payload deflection feedback for vibration reduction. Mech. Syst. Signal Process. 46(1), 70–81 (2014) 15. Sun, N., Fang, Y., Chen, H., Lu, B., Fu, Y.: Slew/translation positioning and swing suppression for 4-DOF tower cranes with parametric uncertainties: design and hardware experimentation. IEEE Trans. Industr. Electron. 63(10), 6407–6418 (2016) 16. Xi, Z., Hesketh, T.: Discrete time integral sliding mode control for overhead crane with uncertainties. IET Control Theory Appl. 4(10), 2071–2081 (2010) 17. Xie, X., Huang, J., Liang, Z.: Vibration reduction for flexible systems by command smoothing. Mech. Syst. Signal Process. 39(1), 461–470 (2013)

Dynamic Modeling and Controller Design of a Novel Aerial Grasping Robot Zhongmou Li1(B) , Xiaoxiao Song1 , Vincent B´egoc1,2 , Abdelhamid Chriette1 , and Isabelle Fantoni1 1

Laboratoire des Sciences du Num´erique de Nantes (LS2N), ECN, UMR CNRS 6004, 44321 Nantes, France {zhongmou.li,xiaoxiao.song,vincent.begoc,abdelhamid.chriette, isabelle.fantoni}@ls2n.fr 2 Institut catholique d’arts et m´etiers (Icam), Lieusaint, France

Abstract. This paper presents dynamic modelling, controller design and simulation results of a novel aerial robot dedicated to manipulation and grasping of large-size objects. This robot can be described as a flying hand composed of four fingers, four quadrotors and a body structure. The four quadrotors are arranged so as to allow the full actuation of the body structure in SE(3). This permits the system to maintain its position and attitude while closing the fingers despite external disturbances. The opening/closing motion of each finger is actuated by the yaw rotation of one quadrotor and transmitted through a non-backdrivable worm-gear mechanism so that the hand can produce secured grasps. We design a model predictive controller to deal with unknown mass, inertia and center of mass. The effectiveness of the controller and its robustness against external disturbances and noise are validated through several simulations using ADAMS-SIMULINK co-simulation.

Keywords: Aerial systems

1

· Mechanics and control · Grasping

Introduction

Unmanned Aerial Vehicles, because of their versatility, are expected to accomplish increasingly complex tasks, such as grasping and manipulating objects [4]. Previous works embed additional mechanisms on quadrotors, such as grippers [2] and/or serial manipulators [10,15]. Adding motors reduces the energetic autonomy, furthermore, it is difficult for those approaches to deal with large objects, as a larger object often requires a larger gripper/manipulator thus reducing the payload. Increasing the payload can be achieved using multiple quadrotors. Quadrotors can be rigidly attached to a manipulated body [8]. However, if all rotors are collinear, the obtained system remains underactuated and does not provide Supported by China Scholarship Council. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 538–546, 2021. https://doi.org/10.1007/978-3-030-58380-4_64

A Novel Aerial Grasping Robot

539

full-actuation in SE(3), such that it cannot resist any lateral disturbance without tilting. Another approach consists of using cable-suspended mechanisms actuated by multiple quadrotors [3,7,13]. However, even if cable robots can provide full actuation, they are not designed to grasp objects and the object needs to be manually attached before performing the task. Another approach to achieve full-actuation in SE(3), consists of using three quadrotors as rotating thrust generators, the generated forces are then transmitted to a frame, either through passive spherical joints [9], or through rigid articulated passive legs [14]. In both cases, the yaw motion of each drone cannot contribute to the manipulation task, making these robots uselessly overactuated. In order to grasp, manipulate and transport large objects, we propose a novel robot architecture, named FlyingGripper, presented in a planar version, in a previous article [12]. The version of the 3D robot can be described as a flying hand composed of a body structure, four fingers and four quadrotors arranged so as to allow the full actuation of the body structure in SE(3) (not all rotors are collinear). This permits the system to maintain its position and attitude while closing the fingers despite any external disturbance. The yaw motion of each quadrotor is used to actuate the opening/closing motion of each finger. A model predictive controller is designed to track a trajectory considering varying dynamic parameters such as model mismatch. Therefore, control input is calculated through a control allocation algorithm. A co-simulation shows the ability of the FlyingGripper to grasp, transport and place a large object in the presence of noise and disturbances. Finally, conclusion and future work are given.

2

Presentation of the FlyingGripper

The FlyingGripper is composed of four quadrotors, four fingers and a body structure Fig. 1. Each finger has two phalanges and is underactuated so that it can adapt to the size and shape of the object and deal with position uncertainty. Each quadrotor is linked to the body structure through a passive revolute joint so that it can rotate along its yaw axis with respect to the body structure. This yaw motion is transmitted through a worm-gear mechanism to an actuation bar to open and close the finger Fig. 1(b). This worm-gear mechanism is chosen to be non-backdrivable to produce form-closure grasps [5]. This stability criteria yields to secured grasps which do not rely on the capability of actuators nor on friction between phalanges and the object, and furthermore do not require any additional energy for grasping during the flight. More details on the closing sequence of the FlyingGripper can be found in [12]. Quadrotors are arranged so as to provide full-actuation of the robot in SE(3), i.e. yaw axes of quadrotors are not colinear but tilted along the x and y directions of the frame FG attached to the body structure Fig. 1. The manipulability of our 3D verison robot is not analyzed in this paper, since a methodology was presented in [12] to optimize the design of a planar version of the FlyingGripper. However, we present here full manipulation of the 3D version of the FlyingGripper, which permits to maintain its position and attitude during grasping despite external disturbances.

540

Z. Li et al.

(a)

(b)

Fig. 1. (a) CAD view of the FlyingGripper composed of four quadrotors, four underactuated fingers and a body structure, (b) Opening/closing motion of each finger does not require any additional actuator since it uses the yaw rotation of each quadrotor transmitted, through a worm-gear mechanism, to the finger’s actuation bar.

3

Static and Dynamic Modeling of the FlyingGripper

T  Let q = ξ T η T γ T be the vector of generalized coordinates which fully describe the configuration of the robot, where: T

– ξ = [x y z] is the position vector of the origin of the body frame FG expressed in the world frame F0 , T – η = [φ θ ψ] is the vector of Euler angles denoting the orientation of FG with respect to F0 , T – γ = [γA γB γC γD ] , with γi the yaw rotation angle of the ith quadrotor expressed in the quadrotor’s local frame Fi (i = A, B, C, D). The following assumptions are made in order to simplify the expressions of the dynamic model: FG coincides with the Center of Mass (CoM) of thewhole system (the robot and the object) and the origin of Fi = Oi , xi , yi , zi , coincides with the CoM of quadrotor i. The static equilibrium of the FlyingGripper is given by: ⎧ D

⎪ G ⎪ Ri fzi zi = 0 ⎨f + i=A (1) D 

⎪ G G ⎪ ⎩τ + ri Ri fzi zi + G Ri τi = 0 i=A

where f and τ are respectively the external force and torque vectors exerted on the body structure expressed in FG , fzi and τi are respectively the thrust force and the torque vector exerted by the ith quadrotor expressed in the local frame Fi , G Ri is the rotation matrix of Fi expressed in FG , G ri is the coordinate ri is the skew-symmetric matrix of G ri . vector of Oi expressed in FG and G  T T T 6×1 Introducing w = f τ ∈R the external wrench applied on the body  T T T T T structure and wq = wA wB wC wD ∈ R16×1 the concatenation of wrenches

A Novel Aerial Grasping Robot

541

T  applied by the four quadrotors on the body structure, with wi = fzi τiT ∈ R4×1 , the equilibrium expression (1) can be written in the following form: w + Wwq = 0 (2)   r r r r WB WC WD ∈ R6×16 is the wrench matrix mapping the where W = WA quadrotors wrenches in the body frame FG . Wir ∈ R6×4 is defined as   G Ri zi 03×3 r Wi = G G . (3) ri Ri z i G Ri The dynamic modeling of the whole system (robot, fingers and object) is developed considering the whole system as a rigid body. The dynamics of the fingers during grasping or releasing is also neglected as the fingers are assumed to close/open slowly. Indeed, we assume that the object, once grasped, has no motion with respect to the body frame. Therefore, the dynamic model is developed by applying Newton-Euler method:  0      RG 0 mI3 03 mg ξ¨ = w (4) + 03 Im ω˙ 0 I3 ω × Im ω where m is the mass and Im is the inertia matrix of the whole manipulator system comprising the quadrotors and the object, ω is the angular velocity of the robot expressed in FG and 0 RG is the rotation matrix of FG relative to F0 .

4

Controller Design

The controller designed for our robot has two objectives: (1) a model predictive controller enables the robot to move from an initial to a final position following a reference trajectory; (2) a control allocation algorithm [11] is developed to ensures the closing/opening of fingers while tracking the trajectory.

Fig. 2. General control scheme of FlyingGripper.

542

4.1

Z. Li et al.

Model Predictive Control for Tracking a Trajectory

For our robot, variations of dynamic parameters caused by holding the object are considered as model mismatch and uncertainty. Then, a model predictive controller based on [1] is developed to track the trajectory, dealing with the varying dynamic parameters, in presence of disturbances and noise. More precisely, Translation MPC and Attitude MPC (Fig. 2) calculate the required forces f and torques τ in order to track the desired trajectory. In the Translation MPC, the input f can be obtained by solving the optimization problem as follows: 

nξ +Np

minimize ξ U

T

[xξd (k) − xξm (k)] Qξ (k) [xξd (k) − xξm (k)]

(5)

k=nξ +1

subject to:

xξm (k) = Aξ xξm (k − 1) + Bξ f (k − 1) + gξ eξ (k) = xξ (k) − xξm (k) xξd (k) = x∗ξ (k) − eξ (k)

(6) (7) (8)

    ξ = f (nξ ), f (nξ + 1), . . . , f (k), . . . , f (nξ + Nc − 1) , while xξ = ξ T ξ˙T T where U is the translation state variable, x∗ξ (k) is the reference translation trajectory. The prediction model in Eq. (6) is linearized from Eq. (4) at the horizontal attitude, xξm is output of the prediction model to predict the robot translation state. eξ is the error between the state xξ and the prediction xξm , while xξd is the correction of the reference trajectory x∗ξ (k) with the error eξ . Noticing that Np is the prediction horizon, Nc is the control horizon and Qξ is the weighting matrix. T  In terms of the Attitude MPC, the state variable is xη = η T η˙ T . The attitude prediction model is based on the linearization of the dynamic model in Eq. (4) at the horizontal attitude to obtain the required torque τ . 4.2

A Control Allocation Algorithm Ensuring Closing/Opening the Fingers

For our robot, the control input of each quadrotor is the motor speed that can be linked to the quadrotor wrench using the following linear relation [6]: wi = Γi ui ,

(9)

where Γi ∈ R4×4 is an invertible matrix whose components are given from aero 2 2 2 2 T dynamic coefficients and geometric parameters, ui = i1 i2 i3 i4 , with ij being the rotational speed of the jth rotor of the ith quadrotor. T  Introducing u = uTA uTB uTC uTD ∈ R16×1 the control input of the robot, Eq. (2) can be rewritten as: w + Au = 0 (10)

A Novel Aerial Grasping Robot

543

with A = WΓ ∈ R6×16 , with Γ = diag (Γi ), is a matrix that maps u ∈ R16×1 onto the wrench w ∈ R6×1 . The control input u must be chosen so as to satisfy Eq. (10) and so that each quadrotor exerts a yaw torque τzi on the worm screw in order to open or close T the corresponding finger. Let introduce τyaw = [τzA τzB τzC τzD ] the vector of yaw torques exerted by the four quadrotors. It can be expressed from the input vector using τyaw = Cu, where C = diag(ci ) with ci being the last row of matrix Γi in Eq. (9). Thus, the control input u can be obtained as     −A w = u = Hu . (11) C τyaw T  T Since the dimension of u ∈ R16×1 is greater than that of wT τyaw ∈ R10×1 , the system is overactuated and there exists an infinity of solutions for u to Eq. (11). The control input u is restricted by the limits of the rotors and must satisfy u ≤ u ≤ u, where u and u are resp. the lower and upper bounds to the rotor square speeds. In order to increase the possibility of finding solutions respecting the motor speed limits, we apply the algorithm in [11] to compute u:    w − Hum (12) u = um + H+ τyaw where um = (u + u) /2 and H+ is the Moore-Penrose pseudoinverse of H.

5

Simulation Results

A simulator was designed using MATLAB-SIMULINK and ADAMS cosimulation. The dynamic of the flying robot was simulated by ADAMS, while the controller was implemented on MATLAB-SIMULINK. A white noise of amplitude ±10−3 (m or rad) was introduced to each measurement of q and a varying disturbance force fd was applied to the CoM of the robot in the ADAMS model. 5.1

Robot Grasping a 1 kg Object with Disturbances

In this first simulation, the robot is required to perform a downward trajectory and the close its fingers while maintaining its position above the object. The results are shown in Fig. 3, where we can see that the robot is able to complete the task with the controller under disturbances and sensor noise. Table 1 gives the mean of absolute value of the tracking error for each coordinate for different amplitudes of the disturbance force.

544

Z. Li et al. Table 1. Mean of absolute value of the tracking error of each coordinate Amplitude of force disturbances (N) 0.1 0.5 1 3 5 Mean error x (m)

4.9e−04 6.1e−04 9.7e−04 2.8e−03 6.3e−03

Mean error y (m)

5.0e−04 6.0e−04 9.5e−04 2.9e−03 6.0e−03

Mean error z (m)

5.1e−03 5.1e−03 5.2e−03 8.4e−03 2.2e−02

Mean error φ (rad) 4.5e−04 4.7e−04 6.0e−04 1.6e−03 8.3e−03 Mean error θ (rad) 4.9e−03 5.0e−03 5.9e−04 2.6e−03 3.2e−03 Mean error ψ (rad) 3.7e−03 3.9e−03 3.9e−04 4.0e−04 5.1e−04

5.2

Robot Grasping, Transporting and Placing a 1 kg Object

In reality, the robot is always required to accomplish more complex tasks, for instance to grasp, manipulate and place an object. Thus, in this simulation, the robot is supposed to approach the object (1 kg), close the fingers, transport the object and place the object under disturbances (Fig. 3) and sensor noise. The simulation results are shown in Fig. 4.

Fig. 3. FlyingGripper performs a downward trajectory (0–2 s) and then closes the fingers while maintaining its position and attitude (2 s–5 s) under noise and a disturbance force

A Novel Aerial Grasping Robot

545

Fig. 4. FlyingGripper performs a downward trajectory approaching the object in 0–3 s, closing the fingers in 3 s–6 s, transporting the object in 6 s–21 s and placing the object in 21 s–25 s under disturbances (Fig. 3) and sensor noise.

This result illustrates that the robot is capable of grasping the object and dealing with the variation of dynamics caused by the object in the manipulation. It demonstrates the effectiveness of the designed controller and its robustness against noise and disturbance. A video of this co-simulation is available online at.1

6

Conclusion and Future Work

In this paper, we have built the static and dynamic model for the novel aerial grasping robot. The developed MPC controller, based on the dynamic model, enables the robot to track a reference trajectory, dealing with varying dynamic parameters, such as unknown mass, inertia and CoM. The control allocation algorithm ensures the closing/opening of the fingers while tracking the trajectory. Simulations results validate effectiveness of the controller and its robustness against external disturbances and noise. The next step is to build the prototype and test the controller in real-time experiments.

References 1. Allibert, G., Courtial, E., Chaumette, F.: Predictive control for constrained imagebased visual servoing. IEEE Trans. Rob. 26(5), 933–939 (2010) 2. Car, M., Ivanovic, A., Orsag, M., Bogdan, S.: Impedance based force control for aerial robot peg-in-hole insertion tasks. In: 2018 IEEE/RSJ IROS, pp. 6734–6739 (2018) 1

https://youtu.be/ziYE SF3t4c.

546

Z. Li et al.

3. Erskine, J., Chriette, A., Caro, S.: Wrench analysis of cable-suspended parallel robots actuated by quadrotor unmanned aerial vehicles. J. Mech. Robot. 11(2), 020909 (2019) 4. Khamseh, H.B., Janabi-Sharifi, F., Abdessameud, A.: Aerial manipulation - a literature survey. Robot. Auton. Syst. 107, 221–235 (2018) 5. Krut, S., B´egoc, V., Dombre, E., Pierrot, F.: Extension of the form-closure property to underactuated hands. IEEE Trans. Rob. 26(5), 853–866 (2010) 6. Mahony, R., Kumar, V., Corke, P.: Multirotor aerial vehicles: modeling, estimation, and control of quadrotor. IEEE Robot. Autom. Mag. 19(3), 20–32 (2012) 7. Manubens, M., Devaurs, D., Ros, L., Cort´es, J.: Motion planning for 6-D manipulation with aerial towed-cable systems. In: Robotics: Science and Systems, Berlin, Germany (May 2013) 8. Michael, N., Fink, J., Kumar, V.: Cooperative manipulation and transportation with aerial robots. Auton. Robot. 30(1), 73–86 (2011) 9. Nguyen, H.N., Park, S., Park, J., Lee, D.: A novel robotic platform for aerial manipulation using quadrotors as rotating thrust generators. IEEE Trans. Rob. 34(2), 353–369 (2018) 10. Orsag, M., Korpela, C., Bogdan, S., Oh, P.: Dexterous aerial robots–mobile manipulation using unmanned aerial systems. IEEE Trans. Rob. 33(6), 1453–1466 (2017) 11. Pott, A., Bruckmann, T., Mikelsons, L.: Closed-form force distribution for parallel wire robots. In: Computational Kinematics, pp. 25–34. Springer (2009) 12. Saint-Sevin, M., B´egoc, V., Briot, S., Chriette, A., Fantoni, I.: Design and optimization of a multi-drone robot for grasping and manipulation of large size objects. In: Proceedings of the 22nd CISM IFToMM Symposium, pp. 458–465 (2018) 13. Sanalitro, D., Savino, H.J., Tognon, M., Cort´es, J., Franchi, A.: Full-pose manipulation control of a cable-suspended load with multiple UAVs under uncertainties. IEEE Robot. Autom. Lett. (2020) 14. Six, D., Briot, S., Chriette, A., Martinet, P.: The kinematics, dynamics and control of a flying parallel robot with three quadrotors. IEEE Robot. Autom. Lett. 3(1), 559–566 (2018) 15. Suarez, A., Soria, P.R., Heredia, G., Arrue, B.C., Ollero, A.: Anthropomorphic, compliant and lightweight dual arm system for aerial manipulation. In: IEEE/RSJ IROS, pp. 992–997 (2017)

Dynamically-Feasible Trajectories for a Cable-Suspended Robot Performing Throwing Operations Deng Lin1 , Giovanni Mottola2 , Marco Carricato2(B) , Xiaoling Jiang1(B) , and Qinchuan Li1 1

Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China [email protected],{jxl1212,lqchuan}@zstu.edu.cn 2 Department of Industrial Engineering, University of Bologna, Bologna, Italy {giovanni.mottola3,marco.carricato}@unibo.it

Abstract. In this paper, we analyze a spatial 3-degree-of-freedom cabledriven robot with a point-mass end-effector suspended by 3 cables. The kinematic and dynamic behaviour of this robot was investigated in previous works, in particular, the possibility of taking advantage of inertia forces on the end-effector to enlarge the robot workspace. Here, we propose to use this highly dynamic robot as a robotic sling, in order to perform throwing operations: the end-effector uses a gripper to carry an object, and the gripper opens at a suitable instant in order to launch the object towards a given target point. The mathematical model and results from numerical simulations are presented. Keywords: Dynamics

1

· Throwing · Cable-suspended · Feasibility

Introduction

This paper studies a cable-suspended parallel robot (CSPR), performing a highly dynamic class of motions. A dynamic trajectory is said to be feasible if all cable tensions τi , i = 1, . . . , n are positive at all times. While feasibility can be verified by integrating the inverse dynamics problem, this method is time-consuming and usually not applicable in real-time scenarios. Therefore, several authors resorted to analytic methods, by usually defining periodic motions where the frequency must lie within a feasible range [5,7,10]. In particular, in [7] the authors defined a class of trajectories as spatial ellipses (on horizontal, vertical or oblique planes), which include straight-line segments and circles as special cases, thus leaving broad freedom in the trajectory planning phase. Dynamic manipulation techniques are not exclusive to CSPRs, but have been successfully applied also to conventional robots with rigid links [6]: other than The work is supported by the National Natural Science Foundation of China (NSFC) under Grants 51525504 and 51805486. c CISM International Centre for Mechanical Sciences 2021  G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 547–555, 2021. https://doi.org/10.1007/978-3-030-58380-4_65

548

D. Lin et al.

enlarging the workspace, they also offer the possibility of greatly reducing operation times and expand the range of achievable tasks (such as juggling and throwing operations). In [9], a 2-DoF robot moves along an elliptical path (in a vertical plane) to periodically throw and catch a ball from the end-effector (EE) by using visual feedback. In [1], the authors propose a casting underactuated 2-DoF planar manipulator whose EE is a gripper attached to a rigid link by a flexible cable; by this design, the authors have shown how target points can be reached even far from the base of the robot, by swinging the rigid link and then releasing the string with the EE at a suitable time. Finally, the concept of casting manipulation was extended to a spatial 3-DoF CSPR in [3]. While this concept allows us to control the EE motion in midair [2], by suddenly pulling on the string, it requires long cables that can be complicated to reel back, especially in cluttered areas. Therefore, in the following we will focus on robots that throw an object from a gripper instead of detaching the gripper itself. Our work is thus closer to [4], where object throwing was proposed for rapid transportation of parts in a manufacturing system; here, however, we consider a more flexible system, where we can freely adjust the position of the throwing point in space by changing the EE trajectory. Applying CSPRs to this goal allows high speeds to be reached by a reduced number of components and limited energy requirements with respect to rigid architectures, as CSPRs have simple mechanical structures [2], limited moving inertias and superior energy efficiency [10]. In particular, we consider a 3-DoF 3-cable spatial CSPR with a point-mass EE (Sect. 2) performing dynamical trajectories along ellipses as in [7,10]. Our results can be directly extended to a CSPR with finite-size EE, provided that simple conditions on the architecture are respected [8]. The goal is to throw an object (modeled as a point mass) from the EE to a target point, by releasing the object at a suitable time. In Sect. 3 we define the ballistic motion of the launched object and find conditions to reach the target point. In Sect. 4 we optimize the trajectory for a given target by a genetic algorithm, in order to minimize the time and energy required; numerical results are also presented. Finally, Sect. 5 presents our conclusions and directions for future work.

2

Dynamic Model

A fixed reference frame with origin O is defined on the robot base (see Fig. 1); in this frame, the fixed cable exit points Ai have position vectors ai . The position of the EE (of mass m) with respect to O is denoted as p = [x, y, z]T . Neglecting cables’ mass and flexibility, the inverse kinematic equations for the cable lengths ρi can be written as  (1) ρi = (p − ai )T (p − ai ), i = 1, 2, 3. The unit vector ei (oriented towards the EE) directed as the ith cable is defined as: (2) ei = (p − ai )/ρi , i = 1, 2, 3.

Dynamically-Feasible Trajectories for a CSPR Performing

549

Fig. 1. A 3-DoF spatial CSPR launching a mass from PL to PT .

The dynamic model of the robot is obtained by writing the force balance of the EE: 3 

(−Fi ei ) + mg = m¨ p

(3)

i=1

Here, Fi is the ith cable tension and g = −[0, 0, g]T is the gravity vector (g is the gravitational acceleration). Combining Eqs. (2) and (3), the equilibrium equation becomes ¨) (4) τ = M−1 (g − p   τ = [τ1 , τ2 , τ3 ]T = m−1 [F1 , F2 , F3 ]T , M = e1 e2 e3 (5) where M is invertible if P does not lie on the plane through the Ai ’s [10]. For the sake of convenience, we assume that P moves along an elliptical trajectory Γ : this way, we can reuse the results from [7], while working with a sufficiently general class of trajectories that appear suitable for throwing motions. The position p is then defined by ⎡ ⎤ ⎡ ⎤   xrs xrc xC sin(ωt) (6) , pC = ⎣ yC ⎦ , b = ⎣ yrs yrc ⎦ p = pC + b cos(ωt) zC zrs zrc pC is the position vector of the trajectory center C, while the entries in b are arbitrary path parameters. The motion amplitude along the x axis is given by xA = x2rs + x2rc , thus the minimum and maximum values of the x coordinate as P moves along Γ are xmin = xC − xA and xmax = xC + xA ; similar limiting values can be found for y and z.

550

3

D. Lin et al.

Launch Trajectory Planning

The target point is pT = [xT , yT , zT ]T (Fig. 1) and the launch instant is tL . By defining αL = ωtL , the launch point pL = [xL , yL , zL ]T and the initial velocity p˙ L = [p˙Lx , p˙Ly , p˙Lz ]T of the launched mass (equal to the velocity of the EE at pL ) are obtained from Eq. (6): pL = pC + b[sin αL , cos αL ]T ,

p˙ L = ωb[cos αL , − sin αL ]T ,

(7)

If we neglect air drag, as in [1–3,6,9], the only external force acting during launch is gravity. So the equation of ballistic flight can be written as: p(t) = g(t − tL )2 /2 + p˙ L (t − tL ) + pL

(8)

whose components along the x, y and z axes are: xT − xL = p˙Lx ΔtLT zT − zL = p˙Lz ΔtLT

yT − yL = p˙Ly ΔtLT 1 − gΔt2LT 2

(9) (10)

where ΔtLT is the time needed to move from PL to PT . From Eqs. (9) we get xT − xL p˙Lx = yT − yL p˙ Ly

(11)

By substituting Eqs. (7) into Eq. (11), we obtain (xT yrs − xC yrs − xrs yT + xrs yC ) cos αL + (−xT yrc + xC yrc + xrc yT − xrc yC ) sin αL + xrs yrc − xrc yrs = 0

(12)

which, using the tangent half-angle formula, can be rewritten as a 2nd -degree equation in tn = tan(αL /2): Gt2n + Htn + I = 0

(13)

where G = xrs yrc − xrc yrs − xT yrs + xC yrs + xrs yT − xrs yC H = 2(−xT yrc + xC yrc + xrc yT − xrc yC ) I = xrs yrc − xrc yrs + xT yrs − xC yrs − xrs yT + xrs yC Solving Eq. (13) yields two values for tn : √ √ −H + H 2 − 4GI −H − H 2 − 4GI , tn,2 = tn,1 = 2G 2G

(14)

and thus two values of αL αL1 = 2 arctan(tn,1 ),

αL2 = 2 arctan(tn,2 )

(15)

Dynamically-Feasible Trajectories for a CSPR Performing

551

By substituting the two values of αLi in Eq. (7), the launch position pL is obtained; we thus find that neither αL nor pL depend on ω. Now, in the expression for p˙ L (Eq. (7)) only ω is unknown. Substituting the components of pL and p˙ L into the first of Eqs. (9) and Eq. (10), we get two nonlinear equations in the two unknowns ω and ΔtLT : xT − xL = ω(xrs cos αLi − xrc sin αLi )ΔtLT zT − zL = ω(zrs cos αLi − zrc sin αLi )ΔtLT −

(16) gΔt2LT /2

(17)

for i = 1, 2. Equation (16) can be rewritten as ω=

xT − xL (xrs cos αLi − xrc sin αLi )ΔtLT

Substituting Eq. (18) into Eq. (17), we find

  2 (xT − xL )(zrs cos αLi − zrc sin αLi ) ΔtLT = − (zT − zL ) g xrs cos αLi − xrc sin αLi

(18)

(19)

Substituting Eq. (19) into Eq. (18), two values of ω can be found. Thus, we obtain the values of ω and ΔtLT that ensure that the launched mass reaches the target point. The method above gives two solutions for αL and thus two solutions for the launch trajectory; it was found that the two values of ω are equal and opposite (at the moment, this is only a conjecture that has been verified for a large number of cases). A necessary and sufficient condition for Eq. (19) to have solutions is (xT − xL ) (zrs cos α − zrc sin α) / (xrs cos α − xrc sin α) − (zT − zL ) > 0 (20) namely (zrs − zrc tan α)/(xrs − xrc tan α) > (zT − zL )/(xT − xL )

(21)

and thus tan α < [(zT − zL )xrs − (xT − xL )zrs ] / [(zT − zL )xrc − (xT − xL )zrc ]

4

(22)

Optimal Trajectory

We now seek an optimal trajectory for the launch motion, given a target point pT ; for the velocity p˙ T = [p˙T x , p˙T y , p˙T z ]T = nT p˙ T  at the target point, we wish to set the direction nT = [nT x , nT y , nT z ]T and a maximum magnitude p˙max (p˙ T  ≤ p˙max ). The velocity p˙ L = [p˙Lx , p˙Ly , p˙Lz ]T at point pL can be calculated as p˙Lx = p˙T x ,

p˙Ly = p˙T y ,

p˙Lz = p˙T z + gΔtLT

(23)

552

D. Lin et al.

For convenience, we define Δ = (xT − xL )2 + (yT − yL )2 and nT,xy = xy,LT  2 2 nT,x + nT yx . Then, from Eqs. (9), the time required to reach the target is ΔtLT = (Δxy,LT )/(p˙ T nT,xy )

(24)

Empirically, we noted that taking into account only ΔtLT in the trajectory design phase leads to solutions with high cable tensions and velocities. As the energy E required to perform the launch motion depends on both these quantities, it is sensible to evaluate it, also to improve energy efficiency. The kinetic energy at the launch point pL is   EK = (1/2)m p˙ 2Lx + p˙ 2Ly + p˙ 2Lz (25) Substituting Eqs. (23) and (24) in Eq. (25) yields, after rearrangement     EK = (1/2)m p˙ T 2 + gΔxy,LT 2 (nT z /nT,xy ) + gΔxy,LT / p˙ T 2 n2T,xy

(26)

The variation of the potential energy from center point PC to launch point PL is ΔEP = mg(zL − zC )

(27)

For simplicity, we choose to use the transition trajectories from [7] to start the EE from rest; by this choice, the robot starts at PC , so the kinetic energy at PC is zero. As the EE moves from PC to PL , the total energy provided by the motors is then at least (28) E = EK + ΔEP A slightly different approach is to minimize, instead of the energy variation at PL with respect to the start condition, the maximum energy variation of the EE as it moves along Γ , as this corresponds to an amount of energy that must have been provided by the robot motors at some time. The total energy can be expressed as ˙ 2 + mg(z − zC ) E = (1/2)mp

(29)

and thus, by differentiating with respect to time E˙ = mp¨ ˙ p + mg z˙ = 0

(30)

˙ and p¨ = ¨ Here, p˙ = p p, while z˙ is the component of p˙ along the z axis. Substituting Eq. (6) in Eq. (30) and introducing s = tan(ωt/2) yields c4 s4 + c3 s3 + c2 s2 + c1 s + c0 = 0

(31)

where c4 = g(zrc − zrs ), c3 = 2ω

2

(x2rs

+

2 yrs

c0 = g(zrc + zrs ) 2 2 2 + zrs − x2rc − yrc − zrc ) − 2gzrc

c2 = 4ω 2 (xrc xrs + yrc yrs + zrc zrs ) − 2gzrc 2 2 2 2 c1 = 2ω 2 (x2rc + yrc + zrc − x2rs − yrs − zrs ) − 2gzrc

Equation (31) is a 4th -degree polynomial equation in the unknown s, which can be solved analytically or numerically; we then get four possible values of α = ωt and, thus, four local extrema of E(α), the largest of which is the maximum energy Emax along Γ.

Dynamically-Feasible Trajectories for a CSPR Performing

553

While any trajectory satisfying the constraints in Sect. 3 is a valid solution, there are in general infinitely many choices; thus, we can choose an optimal solution according to user-defined goals, to increase applicability. As an example, we may wish to minimize both launch time ΔtLT and total energy E, for efficiency. A common approach is to define a single objective function F (such that the optimal choice for pL , pC and b is the one that minimizes F ), as F = βE + γΔtLT

(32)

where β and γ are the weights for E and ΔtLT (β + γ = 1), while E can either be defined as in Eq. (28) or as the maximum energy Emax defined by the second approach. The choice of weights β and γ depends on the relative importance we assign to E and ΔtLT , respectively. As a guideline, β and γ should be chosen so that the first and the second term in Eq. (32) have approximately the same order of magnitude. From Eq. (8), we can prove that, when Δxy,LT is determined, pL can also be found; then, from p˙ T  one can find p˙ L by using Eq. (23). One can see from Eqs. (24), (26), (27) and (29) that, by both approaches, the time to reach PT and the launch energy do not depend on xC and yC . We then assume these coordinates to be given, while we seek to optimize the starting height zC . The optimal path Γopt is uniquely defined from pC , pL , the direction of p˙ L and the linear eccentricity  of Γopt (the distance from the center of the ellipse to either of its two foci). The frequency ω can then be found from Γopt , pL and p˙ L . The optimal choice for Δxy,LT , p˙ T , zC and  is the one that minimizes F . Now we set constraints on the solution by limiting the robot motion within a rectangular cuboid workspace defined as Xmin < x < Xmax , Ymin < y < Ymax , Zmin < z < Zmax . One must also consider the torques and velocities at the actuators: these are equal to the cable tensions (respectively, the cable velocities) times the radius R of the cable winches, so we require max {F1 , F2 , F3 } < Fmax and max {ρ˙ 1 , ρ˙ 2 , ρ˙ 3 } < ρ˙ max at each timestep, where Fmax and ρ˙ max depend on the motors’ properties. Finally, when a CSPR moves along a dynamical trajectory, it must also be ensured that the cable tensions are always positive. For a given Γ , the feasible range [ωmin , ωmax ] for the trajectory frequency ω can be calculated by the method presented in [7]. If (and only if) the value of ω is in this range, then τ1 > 0, τ2 > 0, τ3 > 0, and cable tensions Fi = mτi are positive both before and after launch, since the τi ’s do not depend on the EE mass m. Therefore, the optimization problem also requires the constraint ωmin < ω < ωmax

(33)

We now present a numerical example (time, lengths and forces are given in second, meter and newton, respectively). Given weights β = 0.01, γ = 0.99, parameters pT = [2, 5, −5]T , nT = [1, 4, −8]T and p˙max = 40, we seek the optimal trajectory under constraints Xmin = −4, Xmax = 4, Ymin = −5, Ymax = 5, Zmin = −5, Zmax = 0. For a realistic choice of motors and of radius R, we set Fmax = 8 and ρ˙ max = 7. We also assign the x and y coordinates of the center as xC = −1, yC = 1. To solve the optimization problem, various algorithms could be applied: for instance, we used a genetic algorithm with the MATLAB function ga, setting an initial population of 300 individuals and a maximum of 500 generations. The algorithm converges within the default tolerance limits to a result that we conjecture to be a global optimum, namely ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1.854 −1 −0.674 −2.80 pL = ⎣ 4.42 ⎦ , pC = ⎣ 1 ⎦ , b = ⎣ 0.204 −3.75⎦ , ω = 1.448, min F = 0.303 −4.16 −3.47 −0.993 1.13

554

D. Lin et al.

Figure 2 shows the optimal trajectory and the corresponding cable tensions; notice that the Fi = mτi are discontinuous at the launch instant tL , as the EE mass m suddenly changes.

Fig. 2. Left: 3D view of the trajectory Γopt . Right: the cable tensions along Γopt .

5

Conclusions and Future Work

In this paper, we have considered general elliptical trajectories for a cable-suspended robot with a point-mass end-effector. Then, we designed a trajectory for an object launched from the robot to reach a target point. We derive analytical conditions for the object to reach its target and optimize the motion by using a genetic algorithm. Constraints on the cable tensions are set to guarantee that the trajectories are feasible, both before and after launch. This work is expected to further extend the capabilities of cable robots to perform highly dynamical motions. Directions for further work include: – reducing the discontinuity in the τi ’s (which may cause oscillations after launch); – simulating cable flexibility, especially relevant due to the discontinuity in the τi ’s, as the oscillations could lead (in extreme cases) to losing control of the EE; – finding the Pareto front from a multi-objective optimization, as the objective function in Eq. (32) leaves some arbitrariness in the definition of the weights β and γ; – an experimental validation of our results.

References 1. Arisumi, H., Kotoku, T., Komoriya, K.: A study of casting manipulation (swing motion control and planning of throwing motion). In: Proceedings of the IEEE/RSJ IROS 1997, Grenoble, France, vol. 1, pp. 168–174 (1997). https://doi.org/10.1109/ IROS.1997.649040

Dynamically-Feasible Trajectories for a CSPR Performing

555

2. Arisumi, H., Yokoi, K., Komoriya, K.: Casting manipulation–midair control of a gripper by impulsive force. IEEE Trans. Robot. 24(2), 402–415 (2008). https:// doi.org/10.1109/TRO.2008.918055 3. Fagiolini, A., Torelli, A., Bicchi, A.: Casting robotic end-effectors to reach far objects in space and planetary missions. In: Proceedings of the 9th ESA Workshop on Advanced Space Technologies for Robotics and Automation, ASTRA 2006, Noordwijk, Netherlands (2006) 4. Frank, H., Wellerdick-Wojtasik, N., Hagebeuker, B., Novak, G., Mahlknecht, S.: Throwing objects – a bio-inspired approach for the transportation of parts. In: Proceedings of the IEEE ROBIO 2006, Kunming, China, pp. 91–96 (2006). https:// doi.org/10.1109/ROBIO.2006.340302 5. Jiang, X., Gosselin, C.M.: Dynamically feasible trajectories for three-DOF planar cable-suspended parallel robots. In: Proceedings of the ASME IDETC/CIE 2014, Buffalo, USA (2014). https://doi.org/10.1115/DETC2014-34419 6. Lynch, K.M., Mason, M.T.: Dynamic nonprehensile manipulation: controllability, planning, and experiments. Int. J. Rob. Res. 18(1), 64–92 (1999). https://doi.org/ 10.1177/027836499901800105 7. Mottola, G., Gosselin, C.M., Carricato, M.: Dynamically feasible periodic trajectories for generic spatial three-degree-of-freedom cable-suspended parallel robots. ASME J. Mech. Rob. 10(3), 031004 (2018). https://doi.org/10.1115/1.4039499 8. Mottola, G., Gosselin, C.M., Carricato, M.: Dynamically feasible motions of a class of purely-translational cable-suspended parallel robots. Mech. Mach. Theory 132, 193–206 (2019). https://doi.org/10.1016/j.mechmachtheory.2018.10.017 9. Sakaguchi, T., Fujita, M., Watanabe, H., Miyazaki, F.: Motion planning and control for a robot performer. In: Proceedings of the IEEE ICRA 1993, Atlanta, USA (1993). https://doi.org/10.1109/ROBOT.1993.292262 10. Zhang, N., Shang, W., Cong, S.: Dynamic trajectory planning for a spatial 3-DoF cable-suspended parallel robot. Mech. Mach. Theory 122, 177–196 (2018). https:// doi.org/10.1016/j.mechmachtheory.2015.11.007

Determination of the Geometric Parameters of a Parallel-Serial Rehabilitation Robot Based on Clinical Data Dmitry Malyshev1 , Santhakumar Mohan2 , Larisa Rybak1(B) Gagik Rashoyan3 , and Anna Nozdracheva4

,

1 Belgorod State Technological University named after V.G. Shukhov, Belgorod, Russia

[email protected] 2 Indian Institute of Technology Palakkad, Pudussery East, India 3 Blagonravov Mechanical Engineering Research Institute of the Russian,

Academy of Sciences, Moscow, Russia 4 Gamaleya National Research Center for Epidemiology and Microbiology, Moscow, Russia

Abstract. The article discusses the structure and model of a robotic system for the rehabilitation of the lower limbs based on a passive orthosis in the form of a serial RRRR mechanism and an active parallel 3-PRRR mechanism. Effective numerical methods and algorithms were developed and tested that made it possible to determine the minimum geometric parameters of the active parallel mechanism that ensure the movement of the passive orthosis within the workspace under clinical data when simulating walking. The structure is proposed. The basis parameters for the rehabilitation system design are investigated. To implement the developed methods, an effective algorithm, software package, and visualization system for exported three-dimensional workspaces in STL format were synthesized. The results of mathematical modeling and analysis of the results are given. Keywords: Rehabilitation robot · Geometrical parameters · Optimization · Parallel robot · Workspace

1 Introduction Currently, robotic tools are an optimal way to solve several applied problems in health care. These tasks concern not only the treatment and rehabilitation of patients with impaired musculoskeletal system, but also the implementation of their self-care functions, social adaptation, and replenishment of lost motor and communication functions. It should be noted that the need to create robotic tools to help people with disabilities is steadily growing. This is due to the high frequent disability of the population. Many researchers note that the use of robotic mechanotherapy for patients with acute and chronic periods of stroke can more effectively restore motor function and increase daily activity than using standard rehabilitation methods [1–3]. A study of the effectiveness and safety of the application of the robotic complex Erigo and Lokomat in the acute and early periods of a stroke was carried out. The inclusion of the robotic complex in the © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 556–566, 2021. https://doi.org/10.1007/978-3-030-58380-4_66

Determination of the Geometric Parameters

557

rehabilitation program measures allows activating cortical processes to a greater extent than using only standard methods of rehabilitation treatment and increases the flexibility of the cerebral cortex due to the reorientation of motor neurons of associative fields. It was shown that the inclusion of the locomotor therapy method in the early recovery period of stroke improves biomechanical and clinical neurological parameters and restoration of walking function than using only the traditional rehabilitation complex [4, 5]. The use of rather complicated and expensive medical devices of this type causes the inclusion of mechanotherapy as a class of high-tech devices for clinical practice, its development and production is one of the priority programs for the development of medical equipment. Thus, the development and application in clinical practice of new types of robotic rehabilitation systems with the enhanced structural rigidity of the mechanical structure, ergonomics, and compactness of the system is an urgent and priority area of motor rehabilitation of patients. Lower limb rehabilitation devices are generally having motion in the sagittal plane. Although many therapies need off-planar movements. In specific, hip abduction and adduction are very important therapies. For performing these therapies, only a few attempts have been made as far as lower limb rehabilitation robots are concerned [6, 7]. The device namely Hipbot [6] is designed to give off-planar motions to the lower limbs, when the patient is in lying posture. However, Hipbot has a drawback due to its physical limitations. It can only provide adduction-abduction away from the body and not towards the body. Further, although the system is compact, it is bulky and is di cult to perform treatments in simple manner. Orthoglide [7] based on a spatial three prismaticuniversal-universal (3-PUU) parallel configuration. Even though it has high stiffness, it is associated with smaller singularity-free workspace and complex kinematic relationships. Furthermore, the base (fixed base) of the parallel manipulator is placed on the ground instead of placing it upright or mounted on top. This arrangement further complicated the motions like full knee extension and hip abduction. In order to overcome the above limitations, in this work, one of the popular Cartesian parallel 3-PRRR manipulator [8] is used. In previous works [9, 10], the authors considered a rehabilitation system based on planar parallel 2PRP-2PPR and planar serial RRR manipulators. We developed and tested effective numerical methods and algorithms to determine the optimal geometric parameters of the active manipulator mechanism that provides the required movements inside the workspace. The influence of the range of rotation angles of the joints, which should be provided during the rehabilitation process, on the minimum overall dimensions of the 2PRP-2PPR manipulator is analyzed. These methods have been improved for the new mechanism proposed in this article, which provides spatial movement.

2 Formulation of the Problem In the process of rehabilitation, the human foot and, accordingly, the passive orthosis on which it is fixed, must take many positions under clinical data when simulating walking. The extreme positions of the platform for fixing the foot form the shell of the workspace of the passive orthosis. The active parallel 3-PRRR mechanism determines the position of the platform. It is required to determine its minimum geometric parameters, which

558

D. Malyshev et al.

will provide all the extreme positions of the platform. For knowingly small parameters of the active mechanism and the initial relative position of mechanisms, a quantitative assessment of the unreachable positions set of the platform is determined, which is minimized by changing the relative position of mechanisms using the Hill Climbing technique. In the absence of such a relative position, at which the condition of reachability of all platform positions is satisfied, the geometric parameters of the active mechanism iteratively increase until this condition is met. The conceptual design of a system for lower limb rehabilitation is shown in Fig. 1a. The active 3-PRRR parallel mechanism provides the rehabilitation angles required for rehabilitation of all joints of the patient; a passive orthosis is used to maintain the patient’s lower limb. The structural diagram of a rehabilitation system is presented in Fig. 1b.

Fig. 1. Lower limb rehabilitation system: a - 3D-model of the system, b - structural diagram

An active 3-PRRR parallel robot and a passive orthosis were used in the structure of the rehabilitation system. The active mechanism consists of three kinematic chains Ai Bi Ci Di , i = 3 and has three degrees of freedom - translational movements along each of the axes. The position of the rehabilitation platform, which is an equilateral triangle D1 D2 D3 centered at point P and the radius of the circumscribed circle R, is determined by linear displacements q = (q1 q2 q3 ). We introduce the following notation: ai is the distance between points Ai and Bi , bi is between Bi and Ci , ci is between Ci and Di , di is between Bi and Di , ϕi is the angle of rotation in the joint Ci , θi is the angle of rotation in the joint Bi . A serial RRRR mechanism is used to secure the human foot (Fig. 2). Its joints correspond to the joints of the patient. In the joint E, two movements of the hip joint are provided: rotation in the sagittal plane with angle α and abduction of the leg with angle γ - between the projection of the EF link on the XOY plane and the OY axis. In the knee joint joint F, the link FP is rotated relative to EF by an angle θ . We pose

Determination of the Geometric Parameters

559

the problem of determining the minimum geometric parameters of the active 3-PRRR mechanism under the clinical data for walking.

Fig. 2. Conceptual design of the proposed passive orthosis system

The passive orthosis would benefit the overall rehabilitation system in two ways, one it act as a support system or exoskeleton in order to enhance the limb movements and make sure the proper joints only actuated, and the second one it act as a feedback system through the help of the joint sensors (encoders) presented in the orthosis system. Further, the gravity would affect the overall performance of the physiotherapist due to too much manual effort required during the initial set of therapies. In order to reduce and enhance the physiotherapist efforts, the static balanced passive orthosis is proposed. This proposed system is not only reduce the therapist effort it would also reduce the overall power requirement of the rehabilitation system.

3 A Mathematical Model of a Rehabilitation System Consider the basic kinematic relationships of the active mechanism. Point P is considered as the output link. It coincides with the center of the ankle joint rotary joint. The coordinates of the P point of the output link are determined by the direct kinematics of the active parallel manipulator and is specified in the system of equations ⎧ √ ⎪ ⎨ xP = q1 − 23 R (1) yP = q2 − R2 ⎪ ⎩ zP = q3 The values of the coordinates of the points Di , in turn, are defined as: ⎤ ⎤ ⎡ ⎡ √ √ ⎡ ⎤ xP xP + 23 R xP − 23 R ⎥ ⎥ ⎢ ⎢ D1 = ⎣ yP + R2 ⎦, D2 = ⎣ yP + R2 ⎦, D3 = ⎣ yP − R ⎦, zP zP zP

(2)

560

D. Malyshev et al.

Next, we introduce restrictions on the geometric parameters of the mechanism:

– restrictions on the drive coordinates q: qi ∈ qmin ; qmax : √ q1 = xD1 = xP +

3 R R, q2 = yD2 = yP + , q3 = zD3 = zP . 2 2

(3)

– restrictions on the rotation angles in the joints Ci and Bi : ϕi ∈ [ϕmin ; ϕmax ], θi ∈ [θmin ; θmax ]:  ϕi = cos

−1

 b2 + c2 − di2 , ϕi ∈ [0; π ], 2bc  

(4)

 √ 2 2 where d1 = xP − 23 R + (zP − a)2 , d3 = yP + R2 − a + (zP )2 , d2 =  (xP − a)2 + (yP − R)2 ⎛  ⎞ zP b2 − s12 − s1 (yD1 − a) −1 ⎝ ⎠, θ1 ∈ [0; 2π ], θ1 = cos (5) d1 b  ⎛ ⎞ −xD2 b2 − s22 − s2 (zP − a) ⎠, θ2 ∈ [0; 2π ], (6) θ1 = cos−1 ⎝ d2 b  ⎛ ⎞ −yD3 b2 − s32 − s3 (xD3 − a) ⎠, θ3 ∈ [0; 2π ] (7) θ1 = cos−1 ⎝ d3 b b2 −c2 +d 2

i . where si = 2di The shell of the workspace of the RRRR mechanism is described by a set of threedimensional boxes, which relative to point E have the coordinates:

xPE = xP − xE , yPE = yP − yE , zPE = zP − zE .

(8)

Each of the boxes includes the intervals X PE , Y PE and ZPE . When designing a rehabilitation system, to ensure its compactness, it is important to determine the minimum possible overall dimensions at which it will perform its functions with the given parameters. In this case, these parameters are the dimensions of the drive coordinates qi , the lengths of links bi and ci . The given model takes into account only restrictions on the driving coordinates and the rotation angles in the joints since at this stage the task was to develop and test an approach to determining the optimal geometric parameters of the 3-PRRR mechanism based on a simplified mathematical model that does not take into account gaps in kinematic pairs, real link sizes, drive characteristics, mass of moving parts of an object. In the future, other restrictions will be taken into account with the dynamic model and control system.

Determination of the Geometric Parameters

561

4 Description of Optimization Method Consider the task of determining the minimum geometric parameters of a parallel mechanism, which ensures the attainability of all the provisions of the working platform during rehabilitation. To determine the geometric parameters of the parallel mechanism, it is proposed to use the Hill Climbing technique. It consists of finding a local minimum or maximum by switching between points called nodes, depending on whether the function value is less or more than the current point. This method was considered in more detail in [11]. In [12], it was used to construct an autonomous map and study an unknown environment using mobile robots. The universality of Hill Climbing technique determines its applicability to a wide range of tasks. It can be used to determine the geometric parameters of other mechanisms depending on the required workspace [13]. We chose this approach due to its simplicity and fewer steps to achieve the goal in solving the current problem in comparison with genetic algorithms. In this work, Hill Climbing technique is adapted to the solution of the problem of determining the relative positioning of mechanisms. At various relative positions, the number of boxes from the covering set of the RRRR mechanism, which do not enter the workspace of the active 3-PRRR mechanism, is determined. In the process of performing interval calculations, we will use the methods of interval analysis described in detail in [14]. A precondition for verification is the value of the range of drive coordinates equal to or greater than the overall dimensions of the workspace of the passive mechanism ⎧ ⎨ xPE,max − xPE,min ≤ Q1 (9) y − yPE,min ≤ Q2 ⎩ PE,max zPE,max − zPE,min ≤ Q3     where Qi := Qi , Qi = Qi ≤ qi ≤ Qi are the intervals of the driving coordinates of the active mechanism. In this case, the minimum and maximum values of xPE , yPE and zPE will be determined using the set of boxes that describe the shell of the workspace. If conditions (9) are not satisfied, the active parallel workspace is guaranteed to be insufficient to cover the whole set of positions of the output link during rehabilitation. In this case, the Qi intervals should be increased. We calculate the coordinates of point E, thereby setting the relative position of the active and passive mechanisms. Let’s combine the centers of the described box of the workspace of the RRRR  mechanism with the xE = centerof theranges Qi of the active mechanism:   yE = 

Q2 +Q2

(yPE,max +yPE,min ) , z E 2  −  2 Qi +Qi

=

Q3 +Q3 2

Q1 +Q1 2

x +x − ( PE,max 2 PE,min ) ,

z +z − ( PE,max 2 PE,min ) ,

where is the center of the Qi ranges of the active mechanism, 2  (xPE,max +xPE,min ) , (yPE,max +yPE,min ) , (zPE,max +zPE,min ) is the center of the described box 2

2

2

of the workspace of the RRRR mechanism. If the 3-PRRR mechanism workspace does not include the workspace of the passive one, then it is necessary to check the other relative arrangement of the mechanisms. We introduce additional displacements along

562

D. Malyshev et al.

each of the axes: xE =

  Q1 + Q1 

yE =

zE =

2



−

  xPE,max + xPE,min + x E , 2

2

  yPE,max + yPE,min − + y E , 2

2

  zPE,max + zPE,min − + z E , 2

Q2 + Q2

  Q3 + Q3

(10)

(11)

(12)

where x E , y E , z  E are additional displacements along each axis. For values x E =  E = z E = 0, the center of the described box of the workspace RRRR of the mechanism coincides with the center of the box of the ranges Qi of the active mechanism. When changing the values x E ,y E and z  E , the coordinates of the point E change, therefore, the workspaces of the mechanisms move relative to each other. This allows you to adjust the relative position of the mechanisms to determine the location at which the workspace of the passive mechanism enters the active workspace. For each of the boxes of the workspace of the RRRR mechanism for x E = y E =  z E = 0, it is required to calculate the values of geometric parameters that impose design restrictions on the workspace of the active 3-PRRR mechanism and check for the occurrence of the calculated intervals into specified intervals of geometric parameters: for the intervals X PE , Y PE and ZPE , taking into account formulas (8), (10)–(12), we define the intervals X P , Y P and ZP for which the intervals Qi  of the change in the coordinates are calculated by formulas (3), the intervals i  of the rotation angles in the joints Ci according to the formula (4), the intervals i  of the rotation angles in the joints Bi according to formulas (5)–(7). The system, including the conditions for the occurrence of the calculated intervals in the given, has the form ⎧  ⎨ Qi ⊆ Qi (13)   ⊆ i ⎩ i  i ⊆ i y

Let N be the number of boxes of the workspace of the RRRR mechanism for which system (13) is not satisfied and which do not enter the workspace of the active 3-PRRR mechanism. If N > 0, then to determine the minimum value of N when changing x E , y E , z  E , Hill Climbing technique is used. Starting from the point at x E = y E = z  E = 0 is calculated in each of the neighboring nodes. Denote x E , y E , z  E for the current point as x E,0 , y E,0 , z  E,0 , respectively, and the displacements along each axis to nodes as x E , y E , z  E , which are initially ±Tmax - the maximum distance from the boundary of the workspace of the passive mechanism to the boundaries of the intervals of the driving coordinates of the active mechanism. Figure 3 shows the search for the minimum value of N , with Ni ≤ Ni−1 . If N in another node is less than in the current one, then a transition occurs, that is, x E,0 = x E , y E,0 = y E , z  E,0 = z  E . If in all other nodes N is higher than in the current one, then the displacements x E , y E , z  E decrease to find the minimum of N closer to the

Determination of the Geometric Parameters

563

Fig. 3. Search for the minimum value of N.

current node. The 3 coordinate axes X, Y, Z correspond to the displacements x E , y E , z  E relative to the starting point of the search in the figure. If Nmin > 0, then Qi should be increased until the minimum ranges Qi are found for which N = 0. The algorithm is implemented in the C++ programming language using the Snowgoose interval analysis library [15]. Visualization of the results of modeling the workspace is carried out by converting the list of three-dimensional boxes that describe the workspace into a universal format of 3D models - an STL file.

5 Simulation Results During the simulation, the minimum geometric parameters of the active parallel mechanism were determined for the following sizes: a = 100 mm, qmin = 0 mm, ϕmin = θmin = 10◦ , ϕmax = θmax = 170◦ . The ratio of the lengths of the drive coordinates to the sum of the drive coordinates corresponds to the ratios of the overall dimensions of the passive orthosis of the passive orthosis to the sum of these overall dimensions. This ratio was chosen as a result of a comparative analysis with the case where q1max = q2max = q3max . The dimensions of the links of the active mechanism 3max 3max 2max , b2 = c2 = q1max +q , b3 = c3 = q1max +q . The averare b1 = c1 = q2max +q 4 4 4 q1max +q1max +q1max . The age length of the drive coordinates is calculated as qimax,av = 3 following results were obtained: q1max = 598 mm, q2max = 696 mm, q3max = 1196 mm, xE = 161 mm, yE = 1000 mm, zE = 162 mm. When qimax,av = 830 mm, a mutual arrangement of mechanisms is achieved at which N = 0, therefore, with such dimensions of the drive coordinates, passive orthosis movements within the workspace will be ensured. The change in the amount of N graphically for qimax,av = 820 mm and qimax,av = 830 mm is shown in Fig. 4. To verify the results, the active mechanism workspace was determined under the obtained values of the range of drive coordinates. Figure 5a shows that the passive mechanism workspace is fully included in the active mechanism workspace. Under the obtained results of the geometric parameters of the mechanisms, a rehabilitation system

564

D. Malyshev et al.

Fig. 4. Search for the N minimum value: a - for qimax,av = 820 mm, b - for qimax,av = 830 mm.

3D-model was developed. The movement of the mechanisms was checked under clinical data. Figure 5b shows the development of the trajectory during rehabilitation.

Fig. 5. The workspace of the passive mechanism: a - combined with the workspace of the active mechanism, b - check on the CAD model.

The relative position of the mechanisms and the geometric dimensions of the active 3-PRRR mechanism fully provide the required movements.

6 Conclusion In conclusion, it can be noted that for the proposed lower limb rehabilitation system based on a passive orthosis and an active 3-PRRR parallel robot, effective numerical

Determination of the Geometric Parameters

565

methods and algorithms were developed and tested that made it possible to determine the minimum geometric parameters of the active parallel mechanism. The calculation time for the approximation accuracy δ = 8 mm on a personal computer with an eightcore processor with a frequency of 3,4 GHz was 7 min 35 s. The simulation results were verified by combining with the built-in additional workspace of the active mechanism, as well as on the developed 3D-model of the rehabilitation system. Acknowledgments. This work was supported by the Russian Science Foundation, the agreement number 19-19-00692.

References 1. Cherni, Y., Begon, M., Chababe, H., Moissenet, F.: Use of electromyography to optimize Lokomat® settings for subject-specific gait rehabilitation in post stroke hemiparetic patients: a proof-of-concept study. Neurophysiol. Clin. 47(4), 293–299 (2017) 2. Bruni, M.F., Melegari, C., De Cola, M.C., Bramanti, A., Bramanti, P., Calabrt, R.S.: What does best evidence tell us about robotic gait rehabilitation in strokepatients: A systematic review and meta-analysis. J. Clin. Neurosci. 48, 11–17 (2018) 3. Calabrò, R.S., Naro, A., Russo, M., et al.: Shaping neuroplasticity by using powered exoskeletons in patients with stroke: a randomized clinical trial. J. Neuro. Eng. Rehabil. 15, 35 (2018) 4. Daminov, V.D.: Robotizirovannaja lokomotornaja terapija v nejroreabilitacii. Vestnik vosstanovit Med. 1, 54–59 (2012). (In Russian) 5. Kotsyubinskaya, YuV., Musin, R.R., Safonova, NYu.: The contribution of the robotic complex to the rehabilitation of movements in patients in the early recovery period after cerebral stroke. Neurol. Siberia 2, 34–40 (2018) 6. Guzman-Valdivia, C.H., Blanco-Ortega, A., Oliver-Salazar, M.A., Gomez-Becerra, F.A., Carrera-Escobedo, J.L.: HipBot-the design, development and control of a therapeutic robot for hip rehabilitation. Mechatronics 30, 55–64 (2015) 7. Mohanta, J.K., Mohan, S., Wenger, P., Chevallereau, C.: A new sitting-type lower-limb rehabilitation robot based on a spatial parallel kinematic machine. In: Proceedings of the Asian Mechanism and Machine Science, Bengaluru, India (2018) 8. Gosselin, C.: Compact dynamic models for the Tripteron and Quadrupteron parallel manipulators. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 223(1), 1–12 (2009) 9. Mohan, S., Mohanta, J.K., Behera, L., Rybak, L.A., Malyshev, D.I.: Robust operational-space motion control of a sitting-type lower limb rehabilitation robot. Adv. Intell. Syst. Comput. 1126, 161–172 (2020) 10. Mohan, S., Mohanta, J.K., Kurtenbach, S., Paris, J., Corves, B., Huesing, M.: Design, development and control of a 2PRP-2PPR planar parallel manipulator for lower limb rehabilitation therapies. Mech. Mach. Theor. 112, 272–294 (2017) 11. Mitchell, M.: An Introduction to Genetic Algorithms. The MIT Press, Cambridge (1997) 12. Rocha, R., Ferreira, F., Dias, J.: Multi-robot complete exploration using hill climbing and topological recovery. In: Proceedings of IEEE International Conference on Intelligent Robots and Systems, Nice, France, pp. 1884–1889 (2008) 13. Laryushkin, P., Glazunov, V., Erastova, K.: On the maximization of joint velocities and generalized reactions in the workspace and singularity analysis of parallel mechanisms. Robotica 37(4), 675–690 (2019)

566

D. Malyshev et al.

14. Jaulin, L.: Applied Interval Analysis: with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, New York (2001) 15. Posypkin, M., Usov, A.: Basic numerical routines. https://github.com/mposypkin/snowgoose. Accessed 20 May 2020

Optimal Selection of Transmission Ratio and Stiffness for Series-Elastic Actuators with Known Output Load Torque and Motion Trajectories Guido Bocchieri1,2 , Luca Luzi1 , Nicola Pedrocchi2 , Vincenzo Parenti Castelli1 , and Rocco Vertechy1,2(B) 1 Department of Industrial Engineering, University of Bologna, Bologna, Italy

[email protected] 2 STIIMA Institute, National Research Council of Italy, Milan, Italy

Abstract. Series Elastic Actuators (SEAs) are multi-degree-of-freedom springmass-damper systems that employ elastic transmission elements to reduce mechanical impedance and buffer energy from motor (input) to load (output). This work deals with the mathematical modeling of SEAs and their sizing for given application requirements. In designing traditional actuation systems a common approach is to optimize the speed ratio of the transmission with respect to a given figure of merit, such as the motor rated torque or power, once the output load torque and motion trajectories are given. Taking inspiration from existing techniques developed for the sizing of traditional actuators having just one degree of freedom, a new approach is presented here for two degrees of freedom SEAs, which enables the designer to optimally select at the same time both the ratio and the compliance of the elastic transmission. Keywords: Series elastic actuators · Motor load coupling · Optimization

1 Introduction Series Elastic Actuators (SEAs) have become ubiquitous in the robotics literature since their first appearance in 1995 [1], due to their ability to passively store mechanical energy, thereby providing lower output impedance and tolerance to shocks and impacts. Traditional stiff actuators, despite being perfectly suited for precise and repeatable positioning tasks, fail in applications in which the interaction with the external environment is either necessary, such as in rehabilitation devices where the machine is in physical contact with the patient, or desirable, such as in industrial scenarios where machine operation requires the highest level of safety and compliance to avoid damages to equipment and injuries to people. The introduction of elastic elements on the one hand provides the additional compliance required by the tasks, but on the other hand increases the complexity of the actuator, in terms of both control law development and mechanical design. In this context, proper © CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 567–574, 2021. https://doi.org/10.1007/978-3-030-58380-4_67

568

G. Bocchieri et al.

techniques and guidelines for SEA design need to be devised, possibly adapting those already used for traditional stiff actuators. In the literature, the optimal selection of actuator transmission ratio is a well know problem, for which many different approaches and objective functions have been proposed. As an example, in [2] two decoupled performance indices are defined, one related with the motor and the other with the load characteristics and compared to assess the correct choice of the actuator for the given task. In [3] also the mechanical limits of both motor and transmission are taken into consideration. In [4] the general principle of inertia matching is conceived, which states that under given loading conditions the optimal transmission ratio is the one that balances the inertia of the motor and the load, both referred to the same shaft. Under this condition, the power provided by the motor is therefore equally distributed between motor and load. Detailed practical procedures for the optimal selection of actuator components can be found in [5] and [6]. In [7] this principle is generalized, performing optimization under general load cases and introducing as objective function also the bandwidth of the closed-loop actuator together with energy. In [8] and [9] also the electric motor is taken into consideration, introducing in the optimization problem the power losses in both electrical and mechanical domains. In [10] the problem of the simultaneous optimization of actuator parameters, control efficiency and motion law selection, applied to a manipulator with several motors, is formulated and solved employing multi-objective genetic algorithms. Nonetheless, the vast majority of the existing works only deal with traditional stiff actuators, which are always represented as single degree of freedom mechanical systems. In regard to the modeling of actuators with elastic transmission elements, a very concise and general discussion can be found in [11], which however is not aimed at actuator sizing, while optimization of the natural frequencies arising from the introduction of compliant elements in the power transmission is addressed in [9]. In this work, the combined effect of transmission ratio and stiffness on SEA motor requirements is investigated via a theoretical approach. In particular, a procedure is presented to help the designer in the selection of the best combination of the two parameters that minimize a certain norm function of motor torque and power. The effectiveness of the proposed procedure is proved via a numerical case study.

2 SEA Model 2.1 Description A rotary SEA made by a cascade connection of a motor, a Harmonic Drive (HD) strain wave gear (with circular spline (CS) fixed, wave generator (WG) input and flexible spline (FS) output), a torsional spring and an output link can be modelled as shown in Fig. 1, where: Jm is the motor inertia (it includes the inertias of WG and of all the moving parts that are placed before it); Jg is the FS inertia (it includes the inertias of all parts attached to it and placed between FS and torsional spring); Jl is the load shaft inertia (it includes the inertia of all the moving parts that are placed after the torsional spring); i is the HD gear ratio; k is the spring torsional stiffness; θm , θg and θl are the angular positions of Jm , Jg and Jl ; Cm is the (input) motor torque applied to Jm ; Cext is the (output) external

Optimal Selection of Transmission Ratio and Stiffness

569

load torque applied to Jl . For the sake of simplicity, this model neglects mechanical dissipations and considers a massless torsional spring.

Fig. 1. SEA model.

The system has two degrees of freedom and one control input, Cm . Here, θm and θl are taken as the independent generalized coordinates; owing to HD kinematics iθg = −θm

(1)

2.2 Dynamic Model The study of the effect of transmission ratio and stiffness on SEA motor torque and power for known output load operational specifications requires a dynamic model. Starting from the total kinetic energy, T, and the total potential energy, U,   T = Jm θ˙m2 + Jg θ˙g2 + Jl θ˙l2 2 (2)  2  U = k θl − θg 2

(3)

the following two equations of motions in terms of the generalized coordinates, θm and θl , can be derived via a Lagrangian approach         θ¨m Jm + Jg i2 +  θm i+ θl k i = Cm . (4) θ¨l Jl + θm i + θl k = Cext Equation (4) can be manipulated to express Cm as a function of output load torque and motion trajectories, θl (t) and Cext (t), and transmission parameters i and k; namely, Cm (θl (t), Cext (t), i, k). First, Eq. (4) can be solved for θm , yielding   θm = Cext − Jl θ¨l i k − iθl . (5) Then, substitution of Eq. (5) and of its second time-derivative into the first relation of Eq. (4) provides    

  ....  Cm (θl (t), Cext (t), i, k) = i Jm + Jg i2 C¨ ext − Jl θ l k − θ¨l + Cext − Jl θ¨l i. (6)

570

G. Bocchieri et al.

The respective relation for the required motor power, Pm , can be obtained from Eq. (6) and the first time-derivative of Eq. (5) as follows Pm = Cm θ˙m

(7)

whose explicit expression is complicated, thus not reported here due to space limitation.

3 Optimization Optimal sizing of the SEA schematized in Fig. 1 consists in finding the transmission parameters i and k that minimize a certain p-norm of motor torque or power for known time profiles of θl and Cext within a given period of time ta . The p-norm over ta of a function F(θl (t), Cext (t), i, k), which can be either C m given by Eq. (6) or Pm given by Eq. (7), is defined as: 

F(i, k)p =

ta

p

[F(θl (t), Cext (t), i, k)]p

ta .

(8)

0

Once a selected norm is computed via Eq. (8), determination of optimal transmission parameters i and k can then be done by solving the following system of equations   ∂F(i, k)p ∂i = 0  . (9) ∂F(i, k)p ∂k = 0 Among the possible norms, the most meaningful ones for the definition of motor requirements are those in which p = 2 or p = ∞ [9], corresponding to the root mean square (RMS) and maximum (MAX) values of the function. Here, due to space limits, only the RMS values are considered; namely: Cm, RMS for torque and Pm, RMS for power. Since Eq. (8) depends on the time evolution of θl and Cext within the considered period t a , different sets of optimal transmission parameters exist for different imposed output load operating conditions. As an example, assuming a SEA operating under zero output load torque and with a cycloidal output load motion trajectory, namely    

(10) Cext = 0 and θl (t) =  t ta − sin 2π t ta 2π , solution of Eq. (9) with p = 2 provides the following optimal results: • Minimization of the root mean square value of required motor torque  

      iopt (k) =  Jg + Jl kta2 − 4Jg Jl π 2 Jm kta2 − 4Jl π 2  Cm,RMS



⎧ ⎨0

if k ∗∗ < k < k ∗ iopt , k = 4π  ] ( )[( ) ⎩ otherwise     with k ∗∗ = 4Jg Jl π 2 Jg + Jl ta2 and k ∗ = 4Jl π 2 ta2 

√

Jm kta2 −4Jl π 2 Jg +Jl √ kta2 2

kta2 −4Jg Jl π 2

(11)

(12)

Optimal Selection of Transmission Ratio and Stiffness

571

• Minimization of the root mean square value of required motor power  iopt (k) =

 

  

Jg + Jl kta2 − 4Jg Jl π 2 Jm kta2 − 4Jl π 2 if k ∗∗ < k < k ∗   Pm,RMS i = iopt , k = 0 if k ∗∗ < k < k ∗

(13) (14)

No minimum exists for Pm, RMS in the ranges k ≤ k ∗∗ and k ≥ k ∗ . Note that the possible zeroing of motor RMS torque and power within a defined range of stiffness values is peculiar of the considered special null torque and cycloidal motion trajectory conditions imposed at the SEA output.

4 Numerical Case Study To investigate the sensitivity of motor requirements on SEA transmission ratio and stiffness, and to highlight the effectiveness of the optimization approach described in the previous section, a numerical case study is presented. The example considers the selection of transmission ratio and stiffness for two SEAs with identical inertia parameters but subjected to different output conditions. The considered inertia parameters are: Jm = 3.7515 · 10−5 Kgm2 , Jg = 5.071 · −4 10 Kgm2 , Jl = 8.95 · 10−2 Kgm2 . The first considered output condition consists in zero load torque and cycloidal load motion trajectory, as described by Eq. (10). The second considered output condition consists in zero load torque and Freudenstein 1–3 load motion trajectory, namely   

    Cext = 0 and θl (t) =  t ta − 27 sin 2π t ta 56π − sin 6π t ta 168π . (15) For both trajectories, ta = 1 s and Θ = 3 rad. For the two distinct load cases, the dependencies of required RMS motor torque and power on the ratio and stiffness of the SEA transmission are reported in Figs. 2, 3, 4 and 5. Figures (2.a) and (3.a) respectively show the surface plots of Cm, RMS and Pm, RMS as functions of i and k in the case of cycloidal load motion trajectory, along with the curves (depicted in red solid lines) that represent the zeros and the optima given by Eqs. (11)–(14). As predicted by Eqs. (11)–(14): required RMS motor torque and power can be zero for k ∗∗ < k < k ∗ (here, k ∗∗ ≈ 0.019 Nm/rad and k ∗ ≈ 3.53 Nm/rad); for any fixed value of k in one of the ranges k ≤ k ∗∗ and k ≥ k ∗ there always exists an optimum value for the transmission ratio, iopt (k), that minimizes Cm, RMS , while no such an optimum exists for Pm, RMS . To better highlight the dependencies of motor requirements on transmission parameters, Figs. (2.b) and (3.b) respectively report Cm, RMS and Pm, RMS as functions of i for three values of k: 0.9 Nm/rad and 3.2 Nm/rad, for which both RMS torque and power admit a zero, and 100 Nm/rad, which is a value found in practical SEA implementations;

572

G. Bocchieri et al.

Fig. 2. Cm, RMS vs. i and k in the case of load zero torque and cycloidal motion trajectory.

Fig. 3. Pm, RMS vs. i and k in the case of load zero torque and cycloidal motion trajectory.

whereas Figs. (2.c) and (3.c) respectively report Cm, RMS and Pm, RMS as functions of k for three values of i: 50, which is a value found in practical SEA implementations with HD transmissions, 16 and 5, which are values realizable with one/two-stage planetary gearboxes (the latter being also close to the direct-drive transmission condition, that occurs for i = 1). Comparison of the plots in Fig. 2 to those of Fig. 3 highlights that minimizations of torque and power represent conflicting requirements, especially for the selection of the most suitable transmission ratio. Decision to privilege the former or the latter should be based on application needs. Figures (4.a) and (5.a) respectively show the surface plots of Cm, RMS and Pm, RMS as functions of i and k in thecase of Freudestein 1–3 load  motiontrajectory, along with the  optimum curves Cm, RMS i = iopt , k and Pm, RMS i = iopt , k , depicted with solid red

Optimal Selection of Transmission Ratio and Stiffness

573

Fig. 4. Cm, RMS vs. i and k in the case of load zero torque and Freudestein 1–3 trajectory.

Fig. 5. Pm, RMS vs. i and k in the case of load zero torque and Freudestein 1–3 trajectory.

lines, that can be obtained by following the same procedure described in the previous section to find Eqs. (11)–(14). As shown: no zero exists in this case for the required RMS motor torque and power; a global interior minimum exists for the RMS torque, Cm, RMS ≈ 0.03 Nm, which occurs for k ≈ 3.4 Nm/rad and i ≈ 50; a global interior minimum exists for the RMS power, Pm, RMS ≈ 2.6 W, which occurs for k ≈ 0.9 Nm/rad and i ≈ 16; for any fixed value of k there always exists an optimum value of transmission ratio, iopt (k), that minimizes Cm, RMS , whereas interior optima exist for Pm, RMS only for small values of k. The different shapes of Cm, RMS and Pm, RMS reported in Fig. (2) versus Fig. (4) and in Fig. (3) versus Fig. (5) underline the influence of the load trajectory profile on both motor requirements and optimal transmission parameters. In particular, comparison of

574

G. Bocchieri et al.

Figs. (4.b and c) and (5.b and c) with Figs. (2.b and c) and (3.b and c) highlights that motor torque and power are significantly affected by the chosen load trajectory whenever the transmission is required to be compliant (namely, for low-to-medium values of k).

5 Conclusion This paper investigated the effect of transmission ratio and stiffness on the requirements of motors to be used in series elastic actuators (SEAs). An analytical procedure has been described which enables to find the optimal transmission parameters that minimize motor specifications, either torque or power, for given SEA output load torque and motion trajectories. Plots generated with the procedure can be used by the designer to best select SEA transmission ratio and stiffness based on application needs. A numerical case study has been presented to both show the effectiveness of the proposed procedure and to highlight the influence of imposed output load conditions on optimal transmission parameters and minimal motor requirements. In this work only the case with zero external torque has been considered. Extension of the obtained results to general external torque cases, as well as to SEA models that also include mechanical dissipations and springs with non-negligible mass, will be made in the future to bring the proposed approach closer to practical applications.

References 1. Pratt, G.A., Williamson, M.M.: Series elastic actuators. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Pittsburgh, pp. 399–406 (1995) 2. Legnani, G., Tiboni, M., Adamini, R., Tosi D.: Meccanica degli azionamenti – Vol. 1 – Azionamenti elettrici. 2nd edn. Esculapio, Bologna (2010) 3. Giberti, H., Cinquemani, S., Legnani, G.: Effects of transmission mechanical characteristics on the choice of a motor-reducer. Mechatronics 20(5), 604–610 (2010) 4. Pasch, K., Seering, W.: On the drive systems for high performance machines. ASME J. Mech. Transm. Autom. Des. 106(1), 102–108 (1984) 5. Giberti, H., Cinquemani, S., Legnani, G.: A practical approach to the selection of the motorreducer unit in electric drive systems. Mech. Based Des. Struc. 39(3), 303–319 (2011) 6. Van de Straete, H., Degezelle, P., De Shutter, J., Belmans, R.: Servo motor selection criterion for mechatronic application. Trans. Mechatron. 3(1), 43–50 (1998) 7. Rezazadeh, S., Hurst, J.W.: On the optimal selection of motors and transmissions for electromechanical and robotic systems. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Chicago, pp. 4605–4611 (2014) 8. Bartlett, H.L., Lawson, B.E., Goldfarb, M.: Optimal transmission ratio selection for electric motor driven actuators with known output torque and motion trajectories. ASME. J. Dyn. Sys. Meas. Control. 139(10), 101013 (2017) 9. Verstraten, T., Beckerle, P., Furnémont, R., Mathijssen, G., Vanderborght, B., Lefeber, D.: Series and parallel elastic actuation: impact of natural dynamics on power and energy consumption. Mech. Mach. Theory 102, 232–246 (2016) 10. Padilla-García, E., Villar, C., Rodrguez-Ángeles, A., Moreno-Armendáriz, M.: Concurrent optimization on the powertrain of robot manipulators for optimal motor selection and control in a point-to-point trajectory planning. Adv. Mech. Eng. 9(12), 1–16 (2017) 11. Lee, C., Kwak, S., Kwak, J., Oh, S.: Generalization of series elastic actuator configurations and dynamic behavior comparison. Actuators 6(3), 26 (2017)

Author Index

A Abe, Takashige, 76 Abiko, Satoko, 195 al Hajjar, Nadim, 154 Alfeld, Matthias, 136 Amoozandeh Nobaveh, Ali, 247 Ando, Reiji, 521 Angeles, Jorge, 396, 449 Antonov, Anton, 486 Arakelian, Vigen, 372 Atsumi, Nanato, 109 Ayusawa, Ko, 503

Chen, Xiaoshuai, 76 Chen, Xuechao, 24 Chikhaoui, M. Taha, 512 Chriette, Abdelhamid, 204, 213, 538 Colotti, Alessandro, 432 Coronado, Enrique, 364 Corves, Burkhard, 179, 238, 396, 467

B Bai, Yang, 530 Balderas Hill, Rafael, 213 Baron, Luc, 495 Beckermann, Agnes, 297 Bégoc, Vincent, 538 Belzile, Bruno, 449 Bhattu, Arati Ajay, 414 Bissembayev, Kuatbay, 264 Bocchieri, Guido, 567 Bostanov, Bayandy, 170 Botta, Andrea, 288 Briot, Sébastien, 204, 213, 432, 512

E Ebina, Koki, 76 Emanuelli, Davide, 100 Emaru, Takanori, 16 Endo, Nobutsuna, 313 Erastova, Ksenia, 162 Eto, Ryosuke, 1 F Fantoni, Isabelle, 432, 538 Fomin, Alexey, 486 Fujita, Shunsuke, 16 Furumido, Jun, 76

C Carabin, Giovanni, 100 Carbonari, Luca, 288 Carricato, Marco, 512, 547 Cavallone, Paride, 288 Ceccarelli, Marco, 24, 68 Cenedese, Angelo, 432 Chen, Guangcan, 109

G Gallo, Raimondo, 100 Geng, Jing, 372 Gherman, Bogdan, 154 Glazunov, Victor, 486 Goldsztejn, Alexandre, 432 Gualtieri, Luca, 118 Guha, Anirban, 230

D Dey, Bir Bikram, 305 Dosaev, Marat, 380, 441

© CISM International Centre for Mechanical Sciences 2021 G. Venture et al. (Eds.): ROMANSY 2020, CISM 601, pp. 575–578, 2021. https://doi.org/10.1007/978-3-030-58380-4

576 H Harada, Takashi, 405 Harada, Tomohiro, 195 Hashimoto, Kenji, 34, 355 Hashimoto, Kotaro, 458 Hayasaka, Akinori, 272 Herder, Just L., 247 Higashihara, Takanori, 84 Higuchi, Madoka, 76 Huang, Gao, 24 Huang, Qiang, 24, 68 Hüsing, Mathias, 179, 238, 297, 467 Hwang, Shyh-Shin, 441 I Ibrayev, Aidos, 264 Ibrayev, Sayat, 170, 264 Ibrayeva, Arman, 264 Idà, Edoardo, 512 Inoue, Shou, 338 Ishibashi, Keitaro, 330 Ishii, Hiroyuki, 330, 338 Ishkhanyan, Margarita, 380 Ito, Hirokazu, 92 Iwahara, Naoya, 76 Iwatsuki, Nobuyuki, 255 J Jamalov, Nutpulla, 170, 264 Jenkin, Michael, 305 Jiang, Tian-ci, 127 Jiang, Xiaoling, 547 Jomartov, Assylbek, 170, 264 K Kai, Yoshihiro, 92 Kaimov, Ablay, 170 Kamal, Aziz, 264 Karlsson, Christoffer, 387 Katamura, Ryuta, 109 Kawakami, Yasuo, 34, 355 Kelouwani, Sousso, 495 Kida, Kazuki, 338 Kimura, Naoto, 255 Klimchik, Alexander, 476 Klimina, Liubov, 380, 441 Kobayashi, Yukinori, 16 Kogawa, Atsunori, 92 Komizunai, Shunsuke, 76 Komori, Masaharu, 347 Konno, Atsushi, 76

Author Index Koyama, Tomohiro, 272 Koyanagi, Ken’ichi, 84 Kuatova, Moldyr, 170 Kulkarni, Salil, 414 Kunishige, Yuta, 405 Kuo, Chin-Hsing, 280 Kurashima, Yo, 76 Kurosaki, Masahiro, 1 L Laryushkin, Pavel, 162 Lee, Hee-Hyol, 187 Li, Hui, 68 Li, Long, 68 Li, Qinchuan, 547 Li, Zhongmou, 538 Lim, Hun-ok, 34, 355 Lin, Chyi-Yeu, 280 Lin, Deng, 547 Liu, Ying-Chi, 321 Liu, Yunqi, 68 Lokshin, Boris, 441 Luzi, Luca, 567 M Makino, Koji, 60 Malik, Pramod Kumar, 230 Malyshev, Dmitry, 556 Mamedov, Shamil, 476 Martinet, Philippe, 213 Masterova, Anna, 380 Matsuhiro, Ko, 338 Matsuo, Hiroshi, 521 Matsuura, Daisuke, 109, 521 Mazzetto, Fabrizio, 100 Meng, Libo, 24 Merz, Judith U., 238 Mikhel, Stanislav, 476 Mineshita, Hiroki, 355 Mirz, Christian, 396 Mizukami, Hideki, 34 Mizutani, Ryo, 109 Mohan, Santhakumar, 556 Montazerijouybari, Mohammadreza, 495 Mottola, Giovanni, 547 Murai, Akihiko, 458 N Nakamura, Masahiro, 60 Nakamura, Yoshihiko, 43 Nelson, Carl A., 222

Author Index Nguyen, Vu Linh, 280 Nishikata, Hiromitsu, 272 Nozdracheva, Anna, 556 O Ogura, Teppei, 60 Ohnishi, Kengo, 84 Onda, Moeko, 92 Oshima, Toru, 84 Otani, Takuya, 34, 355 P Pang, Kai, 187 Parenti Castelli, Vincenzo, 567 Pashkevich, Anatol, 476 Patra, Keshab, 230 Pedrocchi, Nicola, 567 Pisla, Doina, 154 Plitea, Nicolae, 154 Popov, Dmitry, 476 Pukhova, Elizaveta, 162 Q Quaglia, Giuseppe, 288 R Radaelli, Giuseppe, 247 Radu, Corina, 154 Rashoyan, Gagik, 556 Rauch, Erwin, 118 Ravankar, Ankit A., 16 Reimer, Felix J., 238 Ren, Hongyuan, 458 Richardsson, Kristoffer, 387 Rincon, Liz, 364 Rojas, Rafael A., 118 Ruiz Garcia, Manuel A., 118 Rybak, Larisa, 556 S Saito, Yukio, 84 Sakaguchi, Masanori, 34, 355 Sakuma, Yutaka, 1 Samsonov, Vitaly, 441 Sase, Kazuya, 76 Schmitz, Markus, 467 Selyutskiy, Yury, 380, 441 Seydakhmet, Askar, 170 Shahidi, Amir, 179 Shimizu, Juri, 34 Shimizu, Soya, 503

577 Shinohara, Nobuo, 76 Solis, Jorge, 387 Song, Xiaoxiao, 538 Sugahara, Yusuke, 109, 338, 521 Suzuki, Ikumi, 272 Svinin, Mikhail, 530 T Takanishi, Atsuo, 34, 330, 338, 355 Takeda, Yukio, 109, 297, 321, 396, 521 Tamamoto, Takumi, 84 Tanaka, Eiichiro, 127, 187 Tanaka, Katsuaki, 338 Tanaka, Takayuki, 458 Tanioka, Tetsuya, 92 Tartari, Michele, 52 Temirbekov, Yerbol, 170 Tempel, Philipp, 136 Terada, Hidetsugu, 60 Terakawa, Tatsuro, 347 Thongsookmark, Chanatip, 297 Tomizawa, Tetsuo, 1 Tomori, Hiroki, 272 Tsujita, Teppei, 1, 76 Tucan, Paul, 154 Tuleshov, Amandyk, 170, 264 U Uzsynski, Olaf, 396 V Vaida, Calin, 154 van der Wijk, Volkert, 136 Venture, Gentiane, 364, 503 Vertechy, Rocco, 567 Vidoni, Renato, 100, 118 Visconte, Carmen, 288 W Wang, Xiang, 68 Wehrle, Erich, 118 Wei, Bin, 145 Weidemann, Carlo, 467 Wojtyra, Marek, 423 Woliński, Łukasz, 423 Y Yamada, Shunsuke, 1 Yamada, Soichi, 330

578 Yamamoto, Ko, 43 Yanagi, Jiei, 330 Yanase, Ryo, 43 Yang, Bo-Rong, 187 Yasuhara, Yuko, 92 Yokoyama, Hiroya, 330 Yu, Shuai-Hong, 187 Yu, Zhangguo, 24

Author Index Z Zaccaria, Federico, 512 Zhang, Huayan, 8 Zhang, Lei, 8 Zhang, Tianwei, 8 Zhong, Tingting, 338 Zhu, Minglei, 204 Zielinska, Teresa, 52